Surface acoustic wave resonator, surface acoustic wave oscillator, and electronic instrument

ABSTRACT

A SAW resonator which, using a quartz crystal substrate with Euler angles (−1.5°≦φ≦1.5°, 117°≦θ≦142°, and 42.79°≦|105|≦49.57°, includes an IDT which excites a stop band upper end mode SAW, and grooves hollowed out of the substrate positioned between electrode fingers configuring the IDT, wherein, when the wavelength of the SAW is λ and the depth of the inter-electrode finger grooves is G, λ and G satisfy the relationship of 0.01λ≦G and wherein, when the line occupation rate of the IDT is η, the groove depth G and line occupation rate η satisfy the relationships of −2.0000×G/λ+0.7200≦η≦−2.5000×G/λ+0.7775 provided that 0.0100λ≦G≦0.0500λ, −3.5898×G/λ+0.7995≦η≦−2.5000+G/λ+0.7775 provided that 0.0500λ&lt;G≦0.0695λ.

TECHNICAL FIELD

The present invention relates to a surface acoustic wave resonator, andto a surface acoustic wave oscillator in which the resonator is mounted,and in particular, relates to a type of surface acoustic wave resonatorwherein grooves are provided in a substrate surface, and to a surfaceacoustic wave oscillator in which the resonator is mounted.

BACKGROUND ART

In a surface acoustic wave (SAW) device (for example, a SAW resonator),the effect of a SAW stop band, piezoelectric substrate (for example,quartz crystal substrate) cut angle, IDT (interdigital transducer)formation shape, and the like, on changes in frequency-temperaturecharacteristics is considerable.

For example, a configuration exciting each of a SAW stop band upper endmode and lower end mode, the distribution of standing waves in each ofthe stop band upper end mode and lower end mode, and the like, aredisclosed in JP-A-11-214958.

Also, points for which the SAW stop band upper end mode has betterfrequency-temperature characteristics than the stop band lower end modeare described in JP-A-2006-148622, JP-A-2007-208871, JP-A-2007-267033and JP-A-2002-100959. Then, it is described in JP-A-2006-148622 andJP-A-2007-208871 that, in order to obtain good frequency-temperaturecharacteristics in a SAW device utilizing a Rayleigh wave, as well asadjusting the cut angle of the quartz crystal substrate, the electrodestandardizing film thickness (H/λ) is increased to around 0.1.

Also, it is described in JP-A-2007-267033 that, as well as adjusting thecut angle of the quartz crystal substrate in a SAW device utilizing aRayleigh wave, the electrode standardizing film thickness (H/λ) isincreased by around 0.045 or more.

Also, it is described in JP-A-2002-100959 that, by using a rotatedY-cut, X-propagating quartz crystal substrate, and utilizing the stopband upper end resonance, the frequency-temperature characteristicsimprove more than in the case of using the stop band lower endresonance.

Also, it is described in JP-A-57-5418 and “Manufacturing Conditions andCharacteristics of Grooved SAW Resonators” (Institute of Electronics andCommunication Engineers of Japan technical research report MW82-59(1982)), that grooves (grooves) are provided between the electrodefingers configuring the IDT, and between the conductor stripsconfiguring the reflectors, in a SAW device using an ST cut quartzcrystal substrate. Also, it is described in “Manufacturing Conditionsand Characteristics of Grooved SAW Resonators” that thefrequency-temperature characteristics change depending on the depth ofthe grooves.

Also, in Japanese Patent No. 3,851,336, as well as describing aconfiguration for making a curve indicating the frequency-temperaturecharacteristics a tertiary curve in a SAW device using an LST cut quartzcrystal substrate, it is described that, in a SAW device using aRayleigh wave, it has not been possible to find a cut angle substratehaving the kind of temperature characteristics indicated by a tertiarycurve.

PROBLEMS THAT THE INVENTION IS TO SOLVE

As heretofore described, there is a wide range of elements for improvingfrequency-temperature characteristics, and it is thought that,particularly with a SAW device using a Rayleigh wave, increasing thefilm thickness of the electrodes configuring the IDT is one factorcontributing to the frequency-temperature characteristics. However, theapplicant has found experimentally that on increasing the film thicknessof the electrodes, environmental resistance characteristics, such astemporal change characteristics and temperature and shock resistancecharacteristics, deteriorate. Also, when having the improvement offrequency-temperature characteristics as a principal object, it isnecessary to increase the electrode film thickness, as previouslydescribed, and an accompanying deterioration of temporal changecharacteristics, temperature and shock resistance characteristics, andthe like, is unavoidable. This also applying to the Q value, it isdifficult to realize a higher Q without increasing the electrode filmthickness.

Consequently, problems when providing the surface acoustic waveresonator and surface acoustic wave oscillator of the invention are,firstly, to realize good frequency-temperature characteristics,secondly, to improve the environmental resistance characteristics, andthirdly, to obtain a high Q value.

SUMMARY

The invention, having been contrived in order to solve at least oneportion of the heretofore described problems, can be realized as thefollowing embodiment or application examples.

APPLICATION EXAMPLE 1

A surface acoustic wave resonator provided on a quartz crystal substratewith Euler angles (−1.5°≦φ≦1.5°, 117°≦θ≦142°, and 42.79°≦|Ψ|≦49.57°)includes an IDT which excites a stop band upper end mode surfaceacoustic wave, and inter-electrode finger grooves hollowed out of thesubstrate positioned between electrode fingers configuring the IDT,wherein, when the wavelength of the surface acoustic wave is λ and thedepth of the inter-electrode finger grooves is G, λ and G satisfy therelationship of

[Expression 1]

0.01λ≦G  (1)

and wherein, when the line occupation rate of the IDT is η, theinter-electrode finger groove depth G and line occupation rate η satisfythe relationships of

[Expression 2]

−2.0000×G/λ+0.7200≦η≦−2.5000×G/λ+0.7775

provided that

0.0100λ≦G≦0.05002  (5)

[Expression 3]

−3.5898×G/λ+0.7995≦η≦−2.5000×G/λ+0.7775

provided that

0.0500λ<G≦0.06952

According to the surface acoustic wave resonator with these kinds ofcharacteristic, it is possible to achieve an improvement infrequency-temperature characteristics.

APPLICATION EXAMPLE 2

With the surface acoustic wave resonator according to applicationexample 1, the inter-electrode finger groove depth G satisfies therelationship of

[Expression 4]

0.01λ≦G≦0.0695λ  (3)

According to the surface acoustic wave resonator with these kinds ofcharacteristic, even in the event that the depth G of theinter-electrode finger grooves deviates due to manufacturing error, itis possible to keep the shift of resonance frequency between individualresonators within a correctable range.

APPLICATION EXAMPLE 3

With the surface acoustic wave resonator according to applicationexample 1 or 2, when the electrode film thickness of the IDT is H, Hsatisfies the relationship of

[Expression 5]

0<H≦0.035λ  (7)

According to the surface acoustic wave resonator with these kinds ofcharacteristic, it is possible to realize an exhibition of goodfrequency-temperature characteristics in an operating temperature range.Also, by having these kinds of characteristic, it is possible tosuppress the deterioration of environmental resistance characteristicsaccompanying an increase in the electrode film thickness.

APPLICATION EXAMPLE 4

With the surface acoustic wave resonator according to applicationexample 3, the line occupation rate η satisfies the relationship of

[Expression 6]

η=−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−

135.99×(H/λ)²+5.817×(H/λ)+0.732−

99.99×(G/λ)×(H/λ)  (8)

By fixing η in such a way as to satisfy Equation (8) in the range of theelectrode film thickness in Application Example 3, it is possible tokeep a secondary temperature coefficient substantially within ±0.01ppm/° C.².

APPLICATION EXAMPLE 5

With the surface acoustic wave resonator according to applicationexample 3 or 4, the sum of the inter-electrode finger groove depth G andelectrode film thickness H satisfies the relationship of

[Expression 7]

0.0407λ≦G+H

By fixing the sum of the inter-electrode finger groove depth G andelectrode film thickness H as in the above equation, it is possible toobtain a higher Q value than with the heretofore known surface acousticwave resonator.

APPLICATION EXAMPLE 6

With the surface acoustic wave resonator according to any one ofapplication examples 1 to 5, Ψ and θ satisfy the relationship of

[Expression 8]

Ψ=1.191×10⁻³×θ³−4.490×10⁻¹×θ²+5.646×10¹×θ−2.324×10³±1.0  (3 1)

By manufacturing the surface acoustic wave resonator using a quartzcrystal substrate cut out at cut angles with these kinds ofcharacteristic, it is possible to obtain a surface acoustic waveresonator exhibiting good frequency-temperature characteristics in awide range.

APPLICATION EXAMPLE 7

With the surface acoustic wave resonator according to any one ofapplication examples 1 to 6, when a stop band upper end mode frequencyin the IDT is ft2, a stop band lower end mode frequency in reflectorsdisposed in such a way as to sandwich the IDT in the propagationdirection of the surface acoustic wave is fr1, and a stop band upper endmode frequency of the reflectors is fr2, ft2, fr1, and fr2 satisfy therelationship of

[Expression 9]

fr1<ft2<fr2  (3 2)

By having these kinds of characteristic, a reflection coefficient |Γ| ofthe reflectors at the stop band upper end mode frequency ft2 of the IDTincreases, and a stop band upper end mode surface acoustic wave excitedwith the IDT is reflected by the reflectors to the IDT side with a highreflection coefficient. Then, the stop band upper end mode surfaceacoustic wave energy confinement becomes stronger, and it is possible torealize a low-loss surface acoustic wave resonator.

APPLICATION EXAMPLE 8

With the surface acoustic wave resonator according to any one ofapplication examples 1 to 7, inter-conductor strip grooves are providedbetween conductor strips configuring the reflectors, and the depth ofthe inter-conductor strip grooves is less than that of theinter-electrode finger grooves.

By having these kinds of characteristic, it is possible to make afrequency shift of the stop band of the reflectors to an area higherthan the stop band of the IDT. For this reason, it is possible torealize the relationship of Equation (32).

APPLICATION EXAMPLE 9

A surface acoustic wave oscillator includes the surface acoustic waveresonator according to any one of application examples 1 to 8, and an ICfor driving the IDT.

APPLICATION EXAMPLE 10

An electronic instrument includes the surface acoustic wave resonatoraccording to application example 1 or 2.

APPLICATION EXAMPLE 11

An electronic instrument includes the surface acoustic wave resonatoraccording to application example 3.

APPLICATION EXAMPLE 12

An electronic instrument includes the surface acoustic wave resonatoraccording to application example 4.

APPLICATION EXAMPLE 13

An electronic instrument includes the surface acoustic wave resonatoraccording to application example 5.

APPLICATION EXAMPLE 14

An electronic instrument includes the surface acoustic wave resonatoraccording to application example 6.

APPLICATION EXAMPLE 15

An electronic instrument includes the surface acoustic wave resonatoraccording to application example 7.

APPLICATION EXAMPLE 16

An electronic instrument includes the surface acoustic wave resonatoraccording to application example 8.

APPLICATION EXAMPLE 17

An electronic instrument includes the surface acoustic wave resonatoraccording to application example 9.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1D are diagrams showing a configuration of a SAW deviceaccording to an embodiment, where FIG. 1A is a plan view of theconfiguration, FIG. 1B is a partial enlarged sectional view of a sidesurface, FIG. 1C is a partial enlarged view for describing details ofFIG. 1B, and FIG. 1D, being the partial enlarged view of FIG. 1C, showsa sectional shape of a groove portion conceivable when manufacturing aSAW resonator using a photolithography technique and an etchingtechnique;

FIG. 2 is a diagram showing an orientation of a wafer which is a basematerial of a quartz crystal substrate used in the invention;

FIGS. 3A and 3B are diagrams showing configuration examples of the SAWdevice when employing a tilted type IDT, where FIG. 1A is an example ofan appearance of tilting electrode fingers, making them perpendicular toan X′ axis, and FIG. 1B is an example of the SAW device having an IDT inwhich bus bars connecting the electrode fingers are tilted;

FIG. 4 is a diagram showing a relationship between a stop band upper endmode and lower end mode;

FIG. 5 is a graph showing a relationship between the depth of aninter-electrode finger groove and a frequency fluctuation amount in anoperating temperature range;

FIG. 6 is a diagram showing temperature characteristics in an ST cutquartz crystal substrate;

FIGS. 7A-7D are graphs showing differences in a change in a secondarytemperature coefficient accompanying a change in a line occupation rateη at a stop band upper end mode resonance point and a stop band lowerend mode resonance point, where FIG. 7A is a graph showing adisplacement of a stop band upper end mode secondary temperaturecoefficient β when a groove depth G is 2% λ, FIG. 7B is a graph showinga displacement of a stop band lower end mode secondary temperaturecoefficient β when the groove depth G is 2% λ, FIG. 7C is a graphshowing a displacement of the stop band upper end mode secondarytemperature coefficient β when the groove depth G is 4% λ, and FIG. 7Dis a graph showing a displacement of the stop band lower end modesecondary temperature coefficient β when the groove depth G is 4% λ;

FIGS. 8A-8I are graphs showing a relationship between the lineoccupation rate η and secondary temperature coefficient β when changingthe inter-electrode finger groove depth, with an electrode filmthickness as 0, where FIG. 8A is a graph when the groove depth G is 1%λ, FIG. 8B is when the groove depth G is 1.25% λ, FIG. 8C is when thegroove depth G is 1.5% λ, FIG. 8D is when the groove depth G is 2% λ,FIG. 8E is when the groove depth G is 3% λ, FIG. 8F is when the groovedepth G is 4% λ, FIG. 8G is when the groove depth G is 5% λ, FIG. 8H iswhen the groove depth G is 6% λ, and FIG. 8I is when the groove depth Gis 8% λ;

FIG. 9 is a graph showing a relationship between an inter-electrodefinger groove depth at which the secondary temperature coefficientbecomes 0 and the line occupation rate η, when the electrode filmthickness is 0;

FIGS. 10A-10I are graphs showing a relationship between the lineoccupation rate η and a frequency fluctuation amount ΔF when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0, where FIG. 10A is a graph when the groove depth G is 1%λ, FIG. 10B is when the groove depth G is 1.25% λ, FIG. 10C is when thegroove depth G is 1.5% λ, FIG. 10D is when the groove depth G is 2% λ,FIG. 10E is when the groove depth G is 3% λ, FIG. 10F is when the groovedepth G is 4% λ, FIG. 10G is when the groove depth G is 5% λ, FIG. 10His when the groove depth G is 6% λ, and FIG. 10I is when the groovedepth G is 8% λ;

FIG. 11 is a graph showing a relationship between the inter-electrodefinger groove depth and the frequency fluctuation amount when theinter-electrode finger groove depth deviates by ±0.001λ;

FIGS. 12A-12F are graphs showing a relationship between theinter-electrode finger groove depth at which the secondary temperaturecoefficient becomes 0 and the line occupation rate η, when the electrodefilm thickness is changed, where FIG. 12A is a graph when the electrodefilm thickness is 1% λ, FIG. 12B is when the electrode film thickness is1.5% λ, FIG. 12C is when the electrode film thickness is 2% λ, FIG. 12Dis when the electrode film thickness is 2.5% λ, FIG. 12E is when theelectrode film thickness is 3% λ, and FIG. 12F is when the electrodefilm thickness is 3.5% λ;

FIGS. 13A-13B are diagrams in which relationships between η1 at whichthe secondary temperature coefficient β≈0 (ppm/° C.²) and theinter-electrode finger groove depth for each electrode film thicknessare summarized in graphs, where FIG. 13A shows a relationship betweenthe groove depth G and η1 when changing the electrode film thicknessfrom 1% λ, to 3.5% λ, and FIG. 13B is a diagram proving that an area inwhich |β|≦0.01 (ppm/° C.²) is inside a polygon formed by linking pointsA to H;

FIG. 14 is a diagram showing in an approximate curve a relationshipbetween the inter-electrode finger groove depth and line occupation rateη for electrode film thicknesses from H≈0 to H=0.035λ;

FIGS. 15A-15F are graphs showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.01λ, where FIG. 15A is a graph when the groove depth G is0, FIG. 15B is when the groove depth G is 1% λ, FIG. 15C is when thegroove depth G is 2% λ, FIG. 15D is when the groove depth G is 3% λ,FIG. 15E is when the groove depth G is 4% λ, and FIG. 15F is when thegroove depth G is 5% λ;

FIGS. 16A-16F are graphs showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.015λ, where FIG. 16A is a graph when the groove depth Gis 0, FIG. 16B is when the groove depth G is 1% λ, FIG. 16C is when thegroove depth G is 1.5% λ, FIG. 16D is when the groove depth G is 2.5% λ,FIG. 16E is when the groove depth G is 3.5% λ, and FIG. 16F is when thegroove depth G is 4.5% λ;

FIGS. 17A-17F are graphs showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.022λ, where FIG. 16A is a graph when the groove depth Gis 0, FIG. 16B is when the groove depth G is 1% λ, FIG. 16C is when thegroove depth G is 2% λ, FIG. 16D is when the groove depth G is 3% λ,FIG. 16E is when the groove depth G is 4% λ, and FIG. 16F is when thegroove depth G is 5% λ;

FIGS. 18A-18F are graphs showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.025λ, where FIG. 18A is a graph when the groove depth Gis 0, FIG. 18B is when the groove depth G is 1% λ, FIG. 18C is when thegroove depth G is 1.5% λ, FIG. 18D is when the groove depth G is 2.5% λ,FIG. 18E is when the groove depth G is 3.5% λ, and FIG. 18F is when thegroove depth G is 4.5% λ;

FIGS. 19A-19F are graphs showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.003λ, where (A) is a graph when the groove depth G is 0,FIG. 19B is when the groove depth G is 1% λ, FIG. 19C is when the groovedepth G is 2% λ, FIG. 19D is when the groove depth G is 3% λ, FIG. 19Eis when the groove depth G is 4% λ, and FIG. 19F is when the groovedepth G is 5% λ;

FIGS. 20A-20F are graphs showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when changingthe inter-electrode finger groove depth, with the electrode filmthickness as 0.035λ, where FIG. 20A is a graph when the groove depth Gis 0, FIG. 20B is when the groove depth G is 1% λ, FIG. 20C is when thegroove depth G is 2% λ, FIG. 20D is when the groove depth G is 3% λ,FIG. 20E is when the groove depth G is 4% λ, and FIG. 20F is when thegroove depth G is 5% λ;

FIGS. 21A-21F are graphs showing the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when changing theinter-electrode finger groove depth, with the electrode film thicknessas 0.01λ, where FIG. 21A is a graph when the groove depth G is 0, FIG.21B is when the groove depth G is 1% λ, FIG. 21C is when the groovedepth G is 2% λ, FIG. 21D is when the groove depth G is 3% λ, FIG. 21Eis when the groove depth G is 4% λ, and FIG. 21F is when the groovedepth G is 5% λ;

FIGS. 22A-22F are graphs showing the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when changing theinter-electrode finger groove depth, with the electrode film thicknessas 0.015λ, where FIG. 22A is a graph when the groove depth G is 0, FIG.22B is when the groove depth G is 1% λ, FIG. 22C is when the groovedepth G is 1.5% λ, FIG. 22D is when the groove depth G is 2.5% λ, FIG.22E is when the groove depth G is 3.5% λ, and FIG. 22F is when thegroove depth G is 4.5% λ;

FIGS. 23A-23F are graphs showing the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when changing theinter-electrode finger groove depth, with the electrode film thicknessas 0.02λ, where FIG. 23A is a graph when the groove depth G is 0, FIG.23B is when the groove depth G is 1% λ, (C) when the groove depth G is2% λ, FIG. 23D is when the groove depth G is 3% λ, FIG. 23E is when thegroove depth G is 4% λ, and FIG. 23F is when the groove depth G is 5% λ;

FIGS. 24A-24F are graphs showing the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when changing theinter-electrode finger groove depth, with the electrode film thicknessas 0.025λ, where FIG. 24A is a graph when the groove depth G is 0, FIG.24B is when the groove depth G is 1% λ, FIG. 24C is when the groovedepth G is 1.5% λ, FIG. 24D is when the groove depth G is 2.5% λ, FIG.24E is when the groove depth G is 3.5% λ, and FIG. 24F is when thegroove depth G is 4.5% λ;

FIGS. 25A-25F are graphs showing the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when changing theinter-electrode finger groove depth, with the electrode film thicknessas 0.03λ, where FIG. 25 A is a graph when the groove depth G is 0, FIG.25B is when the groove depth G is 1% λ, FIG. 25C is when the groovedepth G is 2% λ, FIG. 25D is when the groove depth G is 3% λ, FIG. 25Eis when the groove depth G is 4% λ, and FIG. 25F is when the groovedepth G is 5% λ;

FIGS. 26A-26F are graphs showing the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF when changing theinter-electrode finger groove depth, with the electrode film thicknessas 0.035λ, where FIG. 26A is a graph when the groove depth G is 0, FIG.26B is when the groove depth G is 1% λ, FIG. 26C is when the groovedepth G is 2% λ, FIG. 26D is when the groove depth G is 3% λ, FIG. 26Eis when the groove depth G is 4% λ, and FIG. 26F is when the groovedepth G is 5% λ;

FIGS. 27A-27B are diagrams showing a range in which |β|≦0.01 by means ofgraphs showing a relationship between the line occupation rate η andgroove depth G when the electrode film thickness H is 0H≦0.005λ, whereFIG. 27A shows the case of η1, and FIG. 27B is of η2;

FIGS. 28A-28B are diagrams showing a range in which |β|≦0.01 by means ofgraphs showing a relationship between the line occupation rate η andgroove depth G when the electrode film thickness H is 0.005λ<0.010λ,where FIG. 28A shows the case of η1, and FIG. 28B is of η2;

FIGS. 29A-29B are diagrams showing a range in which |β|≦0.01 by means ofgraphs showing a relationship between the line occupation rate η andgroove depth G when the electrode film thickness H is 0.010λ≦H<0.015λ,where FIG. 29A shows the case of η1, and FIG. 29B is of η2;

FIGS. 30A-30B are diagrams showing a range in which |β|≦0.01 by means ofgraphs showing a relationship between the line occupation rate η andgroove depth G when the electrode film thickness H is 0.015λH<0.020λ,where FIG. 30A shows the case of η1, and FIG. 30B is of η2;

FIGS. 31A-31B are diagrams showing a range in which |β|≦0.01 by means ofgraphs showing a relationship between the line occupation rate η andgroove depth G when the electrode film thickness H is 0.020λ≦H<0.025λ,where FIG. 31A shows the case of η1, and FIG. 31B is of η2;

FIGS. 32A-32B are diagrams showing a range in which |η|≦0.01 by means ofgraphs showing a relationship between the line occupation rate η andgroove depth G when the electrode film thickness H is 0.025λ≦H<0.030λ,where FIG. 32A shows the case of η1, and FIG. 32B is of η2;

FIGS. 33A-33B are diagrams showing a range in which |β|≦0.01 by means ofgraphs showing a relationship between the line occupation rate η andgroove depth G when the electrode film thickness H is 0.030λ≦H<0.035λ,where FIG. 33A shows the case of η1, and FIG. 33B is of η2;

FIGS. 34A-34F are graphs showing relationships between theinter-electrode finger groove depth and an Euler angle Ψ when theelectrode film thickness and line occupation rate η (η1: solid line, η2:broken line) are fixed, where FIG. 34A is a graph when the electrodefilm thickness is 1% λ, FIG. 34B is when the electrode film thickness is1.5% λ, FIG. 34C is when the electrode film thickness is 2% λ, FIG. 34Dis when the electrode film thickness is 2.5% λ, FIG. 34E is when theelectrode film thickness is 3% λ, and FIG. 34F is when the electrodefilm thickness is 3.5% λ;

FIG. 35 is a diagram in which the relationships between theinter-electrode finger groove depth G and Euler angle Ψ for eachelectrode film thickness H are summarized in a graph;

FIG. 36 is a graph showing a relationship between an inter-electrodefinger groove depth at which the secondary temperature coefficient β is−0.01 (ppm/° C.²) and the Euler angle Ψ;

FIG. 37 is a graph showing a relationship between an inter-electrodefinger groove depth at which the secondary temperature coefficient β is+0.01 (ppm/° C.²) and the Euler angle Ψ;

FIGS. 38A-38B are graphs showing a range of Ψ which satisfies therequirement of |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0<H 0.005λ, where FIG. 38A is a graph showing a maximumvalue and minimum value of Ψ, and FIG. 38B is a graph of an area of Ψwhich satisfies the requirement of β;

FIGS. 39A-39B are graphs showing a range of Ψ which satisfies therequirement of |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.005λ<H≦0.010λ, where FIG. 39A is a graph showing amaximum value and minimum value of Ψ, and FIG. 39B is a graph of an areaof Ψ which satisfies the requirement of β;

FIGS. 40A-40B are graphs showing a range of Ψ which satisfies therequirement of |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.010λ<H≦0.015λ, where FIG. 40A is a graph showing amaximum value and minimum value of Ψ, and FIG. 40B is a graph of an areaof Ψ which satisfies the requirement of β;

FIGS. 41A-41B are graphs showing a range of Ψ which satisfies therequirement of |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.015λ<H≦0.020λ, where FIG. 41A is a graph showing amaximum value and minimum value of Ψ, and FIG. 41B is a graph of an areaof Ψ which satisfies the requirement of β;

FIGS. 42A-42B are graphs showing a range of Ψ which satisfies therequirement of |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.020λ<H≦0.025λ, where FIG. 42A is a graph showing amaximum value and minimum value of Ψ, and FIG. 42B is a graph of an areaof Ψ which satisfies the requirement of β;

FIGS. 43A-43B are graphs showing a range of Ψ which satisfies therequirement of |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.025λ<H≦0.030λ, where FIG. 43A is a graph showing amaximum value and minimum value of Ψ, and FIG. 43B is a graph of an areaof Ψ which satisfies the requirement of β;

FIGS. 44A-44B] are graphs showing a range of which satisfies therequirement of |β|≦0.01 (ppm/° C.²) when the range of the electrode filmthickness H is 0.030λ<H 0.035λ, where FIG. 44A is a graph showing amaximum value and minimum value of Ψ, and FIG. 44B is a graph of an areaof Ψ which satisfies the requirement of β;

FIG. 45 is a graph showing a relationship between an Euler angle θ andthe secondary temperature coefficient β when the electrode filmthickness is 0.02λ and the inter-electrode finger groove depth 0.04λ;

FIG. 46 is a graph showing a relationship between an Euler angle φ andthe secondary temperature coefficient β;

FIG. 47 is a graph showing a relationship between the Euler angle θ andEuler angle when the frequency-temperature characteristics are good;

FIG. 48 is a diagram showing an example of frequency-temperaturecharacteristic data from four test specimens under conditions at whichthe frequency-temperature characteristics are best;

FIG. 49 is a graph showing a relationship between a level difference,which is the sum of the inter-electrode finger groove depth andelectrode film thickness, and a CI value;

FIG. 50 is a table showing an example of an equivalent circuit constantand static characteristics of the SAW resonator according to theembodiment;

FIG. 51 is an impedance curve data of the SAW resonator according to theembodiment;

FIG. 52 is a graph for comparing a relationship between the leveldifference and a Q value for the heretofore known SAW resonator and arelationship between the level difference and the Q value for the SAWresonator according to the embodiment;

FIG. 53 is a diagram showing SAW reflection characteristics of the IDTand reflectors;

FIG. 54 is a graph showing a relationship between the electrode filmthickness H and frequency fluctuation in a heat cycle test;

FIGS. 55A-55B are diagrams showing a configuration of a SAW oscillatoraccording to the embodiment;

FIGS. 56A-56B are graphs showing frequency-temperature characteristicsof a SAW resonator, where (A) is a graph showing thefrequency-temperature characteristics of the SAW resonator disclosed inJP-A-2006-203408, and (B) is a graph showing a range of thefrequency-temperature characteristics in an essential operatingtemperature range;

FIG. 57 is a graph showing changes in the frequency fluctuation amountin the operating range for a SAW resonator of which the IDT andreflectors are covered in alumina as a protective film; and

FIG. 58 is a graph showing changes in the frequency fluctuation amountin the operating range for a SAW resonator of which the IDT andreflectors are covered in SiO₂ as a protective film.

DETAILED DESCRIPTION

Hereafter, a detailed description will be given, while referring to thedrawings, of an embodiment according to a surface acoustic waveresonator and surface acoustic wave oscillator of the invention.

Firstly, referring to FIG. 1, a description will be given of a firstembodiment according to the surface acoustic wave (SAW) resonator of theinvention. Of FIG. 1, FIG. 1(A) is a plan view of the SAW resonator,FIG. 1(B) is a partial enlarged sectional view, FIG. 1(C) is an enlargedview for describing details of FIGS. 1(B), and FIG. 1(D) is a diagramwhich, relating to the partial enlarged view of FIG. 1(C), is fordescribing an IDT electrode finger line occupation rate η identificationmethod in the event that the sectional shape is not rectangular buttrapezoidal, which is a conceivable sectional shape when the SAWresonator according to the invention is manufactured using aphotolithography technique and an etching technique. It is appropriatethat the line occupation rate η is a proportion occupied by a width L ofa value (L+S), wherein a protrusion width L and a width S of a groove 32are added, at a height from the bottom of the groove 32 which is ½ of(G+H), which is a value wherein a depth (bump height) G of the groove 32and an electrode film thickness H are added.

The SAW resonator 10 according to the embodiment is basically configuredof a quartz crystal substrate 30, IDT 12, and reflectors 20.

FIG. 2 is a diagram showing an orientation of a wafer 1, which is a basematerial of the quartz crystal substrate 30 used in the invention. InFIG. 2, an X axis is an electrical axis of the quartz crystal, a Y axisis a mechanical axis of the quartz crystal, and a Z axis is an opticalaxis of the quartz crystal. The wafer 1 has a surface wherein a surface2 perpendicular to the Y axis is caused to rotate, with the X axis as arotation axis, an angle of θ′ degrees (degrees) in a direction rotatingfrom a +Z axis toward a −Y axis. An axis perpendicular to the rotatedsurface is a Y′ axis, and an axis parallel to the rotated surface, andperpendicular to the X axis, is a Z′ axis. Furthermore, the IDT 12 andreflectors 20 configuring the SAW resonator 10 are disposed along an X′axis wherein the X axis of the quartz crystal is rotated, with the Y′axis as a rotation axis, +Ψ degrees (or −Ψ degrees), with a directionrotating from a +X axis toward a +Z′ axis as positive. The quartzcrystal substrate 30 configuring the SAW resonator 10 is diced by beingcut out of the wafer 1. Although the shape in plan view of the quartzcrystal substrate 30 is not particularly limited, it may, for example,be a rectangle which has short sides parallel to a Z″ axis, wherein theZ′ axis is rotated +Ψ degrees with the Y′ axis as a rotation axis, andhas long sides parallel to the X′ axis. A relationship between θ′ and anEuler angle θ is θ′=θ−90°.

In the embodiment, an in-plane rotation ST cut quartz crystal substrateexpressed by Euler angles. (−1°≦φ≦1°, 117°≦θ≦142°, and 42.79≦|Ψ|≦49.57°is employed as the quartz crystal substrate 30. Herein, a descriptionwill be given of the Euler angles. A substrate expressed by Euler angles(0°, 0°, and 0°) is a Z cut substrate which has a main surfaceperpendicular to the Z axis. Herein, φ of Euler angles (φ, θ, and Ψ),relating to a first rotation of the Z cut substrate, is a first rotationangle, with the Z axis as a rotation axis, and with a direction rotatingfrom the +X axis to a +Y axis side as a positive rotation angle. TheEuler angle θ, relating to a second rotation carried out after the firstrotation of the Z cut substrate, is a second rotation angle, with the Xaxis after the first rotation as a rotation axis, and with a directionrotating from the +Y axis after the first rotation to the +Z axis as apositive rotation angle. The cut surface of a piezoelectric substrate isdetermined by the first rotation angle φ and second rotation angle θ.The Euler angle Ψ, relating to a third rotation carried out after thesecond rotation of the Z cut substrate, is a third rotation angle, withthe Z axis after the second rotation as a rotation axis, and with adirection rotating, from the +X axis after the second rotation to the +Yaxis side after the second rotation as a positive rotation angle. Apropagation direction of the SAW is expressed by the third rotationangle Ψ with respect to the X axis after the second rotation.

The IDT 12 including a pair of pectinate electrodes 14 a and 14 bwherein the base end portions of a plurality of electrode fingers 18 aand 18 b are connected by bus bars 16 a and 16 b respectively, theelectrode finger 18 a configuring one of the pectinate electrodes 14 a,and the electrode finger 18 b configuring the other pectinate electrode14 b, are alternately disposed with a predetermined space between them.Furthermore, the electrode fingers 18 a and 18 b are disposed in such away that the extension direction of the electrode fingers 18 a and 18 bis perpendicular to the X′ axis, which is the propagation direction ofthe surface acoustic wave, as shown in FIG. 1(A). A SAW excited by theSAW resonator 10 configured in this way, being a Rayleigh type (Rayleightype) SAW, has oscillatory displacement elements on both the Y′ axis andX′ axis. Then, by displacing the SAW propagation direction from the Xaxis, which is the crystal axis of the quartz crystal, in this way, itis possible to excite a stop band upper end mode SAW.

Also, furthermore, it is possible to give the SAW resonator 10 accordingto the invention the kinds of shape shown in FIG. 3. That is, even inthe case of applying an IDT which is tilted by a power flow angle(hereafter called a PFA) δ from the X′ axis, as shown in FIG. 3, it ispossible to achieve a high Q by fulfilling the following requirements.FIG. 3(A) being a plan view showing a working example of a tilted typeIDT 12 a, the disposition conformation of the electrode fingers 18 a and18 b in the tilted type IDT 12 a is tilted in such a way that the X′axis, which is the SAW propagation direction determined by the Eulerangles, and the direction of the electrode fingers 18 a and 18 b of thetilted type IDT 12 a are in a perpendicular relationship.

FIG. 3(B) is a plan view showing another working example of the tiltedtype IDT 12 a. In the example, the electrode finger array direction isdisposed tilted with respect to the X′ axis by tilting the bus bars 16 aand 16 b connecting the electrode fingers 18 a and 18 b respectively,but the configuration is such that the X′ axis and the extensiondirection of the electrode fingers 18 a and 18 b are in a perpendicularrelationship, in the same way as in FIG. 3(A).

Whichever kind of tilted type IDT is used, by disposing the electrodefingers in such a way that a direction perpendicular to the X′ axis isthe extension direction of the electrode fingers, as in the workingexamples, it is possible to realize a low-loss SAW resonator, whilemaintaining good temperature characteristics in the invention.

Herein, a description will be given of the relationship between a stopband upper end mode SAW and a lower end mode SAW. In the stop band lowerend mode and upper end mode SAWs formed by the kind of normal IDT 12shown in FIG. 4 (what are shown in FIG. 4 are the electrode fingers 18configuring the IDT 12), the positions of the anti-nodes (or nodes) ofeach standing wave are apart from each other by π/2. FIG. 4 is a diagramshowing the distribution of the stop band upper end mode and lower endmode standing waves in the normal IDT 12.

According to FIG. 4, as heretofore described, the anti-nodes of the stopband lower end mode standing wave shown by the solid line exist in thecentral position of the electrode fingers 18, that is, in the reflectioncenter position, and the nodes of the stop band upper end mode standingwave shown by the dashed-dotted line exist in the reflection centerposition. In this kind of mode in which the nodes exist in the centralpositions between the electrode fingers, it is not possible toefficiently convert the oscillation of the SAW to a charge with theelectrode fingers 18 (18 a and 18 b), and it is often the case that itis not possible to excite or receive the mode as an electrical signal.However, with the method described in the application, by making theEuler angle Ψ other than zero, and displacing the SAW propagationdirection from the X axis, which is the crystal axis of the quartzcrystal, it is possible to shift the standing wave of the stop bandupper end mode to the position of the solid line in FIG. 4, that is, toshift the standing wave anti-nodes of the mode to the central positionof the electrode fingers 18, and it is possible to excite the SAW of thestop band upper end mode.

Also, one pair of the reflectors 20 are provided in such a way as tosandwich the IDT 12 in the SAW propagation direction. As a specificconfiguration example, both ends of each of a plurality of conductorstrips 22, provided parallel to the electrode fingers 18 configuring theIDT 12, are connected.

With an edge reflection type SAW resonator which actively utilizes areflected wave from the SAW propagation direction end face of the quartzcrystal substrate, or a long IDT type SAW resonator which excites theSAW standing wave with the IDT itself by increasing the number of pairsof IDT electrode fingers, the reflectors are not absolutely necessary.

As the material of the electrode film configuring the IDT 12 andreflectors 20 configured in this way, it is possible to use aluminum(Al), or a metal alloy with Al as its base.

By making the electrode thickness of the electrode film configuring theIDT 12 and reflectors 20 extremely small, the effect of the temperaturecharacteristics possessed by the electrodes is kept to a minimum.Furthermore, making the depth of the quartz crystal substrate portiongrooves large, good frequency-temperature characteristics are derivedfrom the performance of the quartz crystal substrate portion grooves,that is, by utilizing the good temperature characteristics of the quartzcrystal. Because of this, it is possible to reduce the effect of theelectrode temperature characteristics on the temperature characteristicsof the SAW resonator and, provided that the fluctuation of the electrodemass is within 10%, it is possible to maintain good temperaturecharacteristics.

For the above-mentioned reasons, in the event of using an alloy as theelectrode film material, the ratio by weight of metals other than themain element aluminum should be 10% or less, and preferably 3% or less.In the event of using electrodes with a metal other than Al as a base,the electrode film thickness should be adjusted so that the mass of theelectrodes is within ±10% of that when using Al. By so doing, goodtemperature characteristics equivalent to those when using Al can beobtained.

The quartz crystal substrate 30 in the SAW resonator 10 with theheretofore described kind of basic configuration is such that thegrooves (inter-electrode finger grooves) 32 are provided between theelectrode fingers of the IDT 12 and between the conductor strips of thereflectors 20.

When the SAW wavelength in the stop band upper end mode is λ, and thegroove depth is G, the groves 32 provided in the quartz crystalsubstrate 30 should be such that

[Expression 10]

0.01λ≦G  (1)

When fixing an upper limit for the groove depth G, it should be withinthe range of

[Expression 11]

0.01λ≦G≦0.094λ  (2)

as can be seen by referring to FIG. 5. This is because, by fixing thegroove depth G within this kind of range, it is possible to keep thefrequency fluctuation amount within the operating temperature range(−40° C. to +85° C.) at or below the target value of 25 ppm, to bedescribed in detail hereafter. Also, it is preferable that the groovedepth G is within the range of

[Expression 12]

0.01λ≦G≦0.0695λ  (3)

By fixing the groove depth G within this kind of range, even in theevent that manufacturing variation occurs in the groove depth G, it ispossible to keep the shift amount of resonance frequency betweenindividual SAW resonators 10 within a correctable range.

Also, the line occupation rate η is a value wherein the line width L ofthe electrode finger 18 (in the case of the quartz crystal protrusiononly, the width of the protrusion) is divided by the pitch λ/2 (=L+S)between the electrode fingers 18, as shown in FIGS. 1C and 1D.Consequently, the line occupation rate η can be expressed by Equation(4).

$\begin{matrix}\left\lbrack {{Expression}{\mspace{14mu} \;}13} \right\rbrack & \; \\{\eta = \frac{L}{L + S}} & (4)\end{matrix}$

Herein, with the SAW resonator 10 according to the embodiment, the lineoccupation rate η should be fixed within the kind of range whichsatisfies Equations (5) and (6). As can also be understood fromExpressions (5) and (6), η can be derived by fixing the depth G of thegrooves 32.

[Expression 14]

−2.0000×G/λ+0.7200≦η≦−2.5000×G/λ+0.7775

provided that

0.01002λ≦G≦0.05002  (5)

[Expression 15]

−3.5898×G/λ+0.7995≦η≦−2.5000×G/λ+0.7775

provided that

0.0500λ<G0.0695λ  (6)

Also, it is preferable that the film thickness of the electrode filmmaterial (IDT 12, reflectors 20, and the like) in the SAW resonator 10according to the embodiment is within the range of

[Expression 16]

0<H≦0.035λ  (7)

Furthermore, when taking into consideration the thickness of theelectrode film shown in Equation (7) with regard to the line occupationrate η, η can be obtained from Equation (8).

[Expression 17]

η=−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−

135.99×(H/λ)²+5.817×(H/λ)+0.732−

99.99×(G/λ)×(H/λ)  (8)

The manufacturing variation of the electrical characteristics(particularly the resonance frequency) being greater the greater theelectrode film thickness, it is highly likely that the manufacturingvariation of the line occupation rate η is ±0.04 or less when theelectrode film thickness H is within the range of Equations (5) and (6),and that a manufacturing variation greater than ±0.04 occurs whenH>0.03λ. However, provided that the electrode film thickness H is withinthe range of Equations (5) and (6), and the variation of the lineoccupation rate η is ±0.04 or less, it is possible to realize a SAWdevice with a low secondary temperature coefficient β. That is, a lineoccupation rate η up to the range of Equation (9), wherein a toleranceof ±0.04 is added to Equation (8), is allowable.

[Expression 18]

η=−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−

135.99×(H/λ)²+5.817×(H/λ)+0.732−

99.99×(G/λ)×(H/λ)±0.04  (9)

With the SAW resonator 10 according to the embodiment with theheretofore described kind of configuration, in the event that thesecondary temperature coefficient β is within ±0.01 (ppm/° C.²), andpreferably the SAW operating temperature range is −40° C. to +85° C.,the object is to improve the frequency-temperature characteristics to adegree whereby it is possible to keep a frequency fluctuation amount ΔFin the Operating temperature range at or under 25 ppm.

Generally, however, the temperature characteristics of a surfaceacoustic wave resonator are expressed by the following equation.

Δf=α×(T−T ₀)+β×(T−T ₀)²

Herein, Δf represents the frequency change amount (ppm) between thetemperature T and the peak temperature T₀, α the primary temperaturecoefficient (ppm/° C.), β the secondary temperature coefficient (ppm/°C.²), T the temperature, and T₀ the temperature at which the frequencyis highest (the peak temperature).

For example, in the event that the piezoelectric substrate is formed ofa so-called ST cut (Euler angles (φ, θ, and Ψ)=(0°, 120° to 130°, and0°) quartz crystal substrate, the primary constant α=0.0, and thesecondary constant β=−0.034, which when expressed in a graph is as inFIG. 6. In FIG. 6, the temperature characteristics describe an upwardlyconvex parabola (quadratic curve).

With the kind of SAW resonator shown in FIG. 6, the frequencyfluctuation amount with respect to the temperature change is extremelylarge, and it is necessary to suppress the frequency change amount Δfwith respect to the temperature change. Consequently, there is a need torealize a surface acoustic wave resonator based on new knowledge, sothat the secondary temperature coefficient β shown in FIG. 6 is broughtcloser to 0, and the frequency change amount Δf with respect to thetemperature (operating temperature) change when the SAW resonator isactually used comes closer to 0.

Consequently, one object of the invention is to solve the heretoforedescribed kind of problem, making the frequency-temperaturecharacteristics of the surface acoustic wave device extremely good ones,and realizing a surface acoustic wave device which operates with astable frequency, even though the temperature changes.

How a solution to the heretofore described kind of problem may berealized with a SAW device to be configured including the heretoforedescribed kind of technical thought (technical components), that is, howthe inventor arrived at the knowledge according to the invention byrepeatedly carrying out simulations and experiments, will be describedin detail and proved hereafter.

With a SAW resonator using the previously described quartz crystalsubstrate called an ST cut, with the propagation direction the crystal Xaxis direction, in the event that the operating temperature range is thesame, the frequency fluctuation amount ΔF in the operating temperaturerange is approximately 133 (ppm), and the secondary temperaturecoefficient β is about −0.034 (ppm/° C.²). Also, in the event ofutilizing a stop band lower end mode excitation in a SAW resonator usingan in-plane rotation ST cut quartz crystal substrate with the sameoperating temperature range, with the quartz crystal substrate cutangles and SAW propagation direction (0, 123°, and 45°) in Euler anglerepresentation, the frequency fluctuation amount ΔF is approximately 63ppm, and the secondary temperature coefficient β is about −0.016 (ppm/°C.²).

The SAW resonators using the ST cut quartz crystal substrate andin-plane rotation ST cut quartz crystal substrate both utilizing surfaceacoustic waves called Rayleigh waves, the variation of frequency andfrequency-temperature characteristics with respect to the machiningaccuracy of the quartz crystal substrate and electrodes is extremelysmall in comparison with the surface acoustic wave, called a Leaky wave,of an LST cut quartz crystal substrate, meaning that they are mostsuitable for mass production, and are used in various kinds of SAWdevice. However, with the SAW resonators using the ST cut quartz crystalsubstrate, in-plane rotation ST cut quartz crystal substrate, or thelike utilized to date, as previously described, the secondarytemperature characteristics being such that the curve indicating thefrequency-temperature characteristics is a quadratic curve, andfurthermore, the absolute value of the secondary temperature coefficientof the secondary temperature characteristics being large, the frequencyfluctuation amount in the operating temperature range is large, and theycannot be utilized in a SAW device such as a resonator or oscillatorused in a wired communication device or wireless communication devicewhich requires a stability of frequency. For example, in the event thatit is possible to obtain frequency-temperature characteristics whichhave secondary temperature characteristics wherein the secondarytemperature coefficient β is ±0.01 (ppm/° C.²) or less, corresponding toan improvement in the ST cut quartz crystal substrate secondarytemperature coefficient β of ⅓ or less, and in the in-plane rotation STcut quartz crystal substrate secondary temperature coefficient β of 37%or more, it is possible to realize a device which requires that kind offrequency stability. Furthermore, in the event that it is possible toobtain tertiary temperature characteristics, wherein the secondarytemperature coefficient β is substantially zero, and the curveindicating the frequency-temperature characteristics is a tertiarycurve, it is more preferable, as the frequency stability in theoperating temperature range further increases. With tertiary temperaturecharacteristics such as these, it is possible to obtain an extremelyhigh frequency stability of ±25 ppm or less, which has not beenrealizable with the heretofore known kind of SAW device, even in thebroad operating temperature range of −40° C. to +85° C.

The fact that the electrode finger 18 line occupation rate η in the IDT12, electrode film thickness H, groove depth G, and the like, arerelated to the change in the frequency-temperature characteristics ofthe SAW resonator 10, as heretofore described, has been made clear byknowledge based on the simulations and experiments carried out by theinventor. Then, the SAW resonator 10 according to the embodimentutilizes the excitation of the stop band upper end mode.

FIGS. 7A to 7D are graphs showing the change in the secondarytemperature coefficient β with respect to the change in the lineoccupation rate η in a case of exciting and propagating a SAW on thesurface of the quartz crystal substrate 30, with the electrode filmthickness H in FIG. 1(C) as zero (H=0% λ), that is, in a condition inwhich the grooves 32 configured of uneven quartz crystal are formed inthe surface of the quartz crystal substrate 30. Of FIG. 7, FIG. 7(A)shows the secondary temperature coefficient β for the stop band upperend mode resonance when the groove depth G is 0.02λ, and FIG. 7(B) showsthe secondary temperature coefficient β for the stop band lower end moderesonance when the groove depth G is 0.02λ. Also, of FIG. 7, FIG. 7(C)shows the secondary temperature coefficient β for the stop band upperend mode resonance when the groove depth G is 0.04λ, and FIG. 7(D) showsthe secondary temperature coefficient β for the stop band lower end moderesonance when the groove depth G is 0.04λ. The simulations shown inFIG. 7 show examples of cases in which a SAW is propagated in some wayin a quartz crystal substrate 30 on which no electrode film is provided,in order to reduce factors causing the frequency-temperaturecharacteristics to fluctuate. Also, Euler angles (0°, 123°, and Ψ) areused for the cut angles of the quartz crystal substrate 30. With regardto Ψ, a value which is the minimum absolute value of the secondarytemperature coefficient β is selected as appropriate.

From FIG. 7, it can be seen that, both in the case of the stop bandupper end mode and in the case of the lower end mode, the secondarytemperature coefficient β changes considerably in the area in which theline occupation rate η is 0.6 to 0.7. Then, when comparing the change inthe secondary temperature coefficient β in the stop band upper end modeand the change in the secondary temperature coefficient β in the stopband lower end mode, the following points can be seen. That is, thechange in the secondary temperature coefficient β in the stop band lowerend mode is such that the characteristics deteriorate due to thesecondary temperature coefficient β changing from the minus side tofarther on the minus side (the absolute value of the secondarytemperature coefficient β increases). As opposed to this, the change inthe secondary temperature coefficient β in the stop band upper end modeis such that the characteristics improve due to the secondarytemperature coefficient β changing from the minus side to the plus side(a point exists at which the absolute value of the secondary temperaturecoefficient β decreases).

From this, it is clear that in order to obtain goodfrequency-temperature characteristics in a SAW device, it is preferableto use the oscillation of the stop band upper end mode.

Next, the inventor investigated the relationship between the lineoccupation rate η and secondary temperature coefficient β whenpropagating a stop band upper end mode SAW in quartz crystal substrateswith variously changed groove depths G.

FIG. 8 is graphs showing evaluation results when simulating therelationship between the line occupation rate η and secondarytemperature coefficient β when changing the groove depth G from 0.01λ(1% λ) to 0.08λ (8% λ), with the electrode film thickness H as zero(H=0% λ), as in FIG. 7. From the evaluation results, it can be seen thata point at which α=0, that is, a point at which the approximate curveindicating the frequency-temperature characteristics describes atertiary curve, begins to appear from around the point at which thegroove depth G is made 0.0125λ (1.25% λ), as shown in FIG. 8(B). Then,it is also clear from FIG. 8 that η at which β=0 exists in two places (apoint (η1) at which β=0 where η is larger, and a point (η2) at which β=0where η is smaller). Furthermore, it can also be seen from theevaluation results shown in FIG. 8 that the fluctuation amount of theline occupation rate η with respect to the change in the groove depth Gis greater at η2 than at η1.

Regarding this point, it is possible to increase an understandingthereof by referring to FIG. 9. FIG. 9 is a graph plotting η1 and η2, atwhich the secondary temperature coefficient β becomes 0, in the case ofchanging the groove depth G. From FIG. 9, it can be seen that η1 and η2both become smaller as the groove depth G increases, but the fluctuationamount of η2 is so great that, on a graph in which the scale of thevertical axis η is shown in a range of 0.5λ to 0.9λ, it goes off thescale around the point at which the groove depth G=0.04λ. That is, itcan be said that the fluctuation amount of η2 with respect to the changein the groove depth G is large.

FIG. 10 is graphs with the electrode film thickness H as zero (H=0% λ),as in FIGS. 7 and 8, and with the vertical axis of FIG. 8 shown as thefrequency fluctuation amount ΔF instead of the secondary temperaturecoefficient β. From FIG. 10, it can of course be seen that the frequencyfluctuation amount ΔF decreases at the two points (η1 and η2) at whichβ=0. Furthermore, it can be seen from FIG. 10 that of the two points atwhich β=0, the frequency fluctuation amount ΔF is kept smaller at thepoint corresponding to η1 in every graph in which the groove depth G ischanged.

According to the heretofore described tendency, it can be supposed thatit is preferable to employ the β=0 point at which the frequencyfluctuation amount with respect to the fluctuation in the groove depth Gis smaller, that is, η1, for a mass production article in whichdiscrepancies are liable to occur when manufacturing. FIG. 5 shows agraph of the relationship between the frequency fluctuation amount ΔF atthe point (η1) at which the secondary temperature coefficient β issmallest and the groove depth G, for each groove depth G. According toFIG. 5, the lower limit value of the groove depth G for which thefrequency fluctuation amount ΔF is the target value of 25 ppm or lessbeing the groove depth G of 0.01λ, the range of the groove depth G isthat or greater, that is, 0.01≦G.

In FIG. 5, examples are also added of cases in which, by simulation, thegroove depth G is 0.08 or more. According to the simulation, thefrequency fluctuation amount ΔF is 25 ppm or less when the groove depthG is 0.01λ or more, and subsequently, the frequency fluctuation amountΔF decreases every time the groove depth G increases. However, when thegroove depth G becomes approximately 0.09λ or more, the frequencyfluctuation amount ΔF increases again, and on the groove depth Gexceeding 0.094λ, the frequency fluctuation amount ΔF exceeds 25 ppm.

The graph shown in FIG. 5 is of a simulation in a condition in which noelectrode film is formed on the IDT 12, reflectors 20, and the like, onthe quartz crystal substrate 30 but, as can be understood by referringto FIGS. 21 to 26, whose details are shown hereafter, it is supposedthat it is possible to reduce the frequency fluctuation amount ΔF morein a SAW resonator 10 on which an electrode film is provided. Therefore,when fixing the upper limit value of the groove depth G, it should bemade the maximum value in the condition in which no electrode film isformed, that is, G≦0.94λ, and it is possible to represent the preferredrange of the groove depth G for achieving the target as

[Expression 19]

0.01λ≦G≦0.094λ  (2)

The groove depth G has a maximum variation of around ±0.0012 in the massproduction process. Therefore, the individual frequency fluctuationamounts Δf of the SAW resonator 10 in a case in which the groove depth Gdeviates by ±0.001λ, when the line occupation rate η is taken to be aconstant, are shown in FIG. 11. According to FIG. 11, it can be seenthat in a case in which the groove depth G deviates by ±0.001λ whenG=0.04λ, that is, when the groove depth is in a range of0.039λ≦G≦0.041λ, the frequency fluctuation amount Δf is around ±500 ppm.

Herein, in the event that the frequency fluctuation amount Δf is lessthan ±1000 ppm, frequency adjustment is possible using various frequencyfine adjustment methods. However, in the event that the frequencyfluctuation amount Δf is ±1000 ppm or more, adjusting the frequency hasan effect on static characteristics such as the Q value and CI (crystalimpedance) value, and on long-term reliability, leading to a reductionin the yield rate as the SAW resonator 10.

By deriving an approximate equation indicating the relationship betweenthe frequency fluctuation amount Δf (ppm) and groove depth G for thestraight line linking the plots shown in FIG. 11, it is possible toobtain Equation (10).

[Expression 20]

Δf=16334(G/λ)−137  (1 0)

Herein, on calculating the values of G at which Δf<1000 ppm, it is foundthat G≦0.0695λ. Consequently, it can be said that it is preferable thatthe range of the groove depth G according to the embodiment is optimally

[Expression 21]

0.01λ≦G≦0.0695λ  (3)

Next, FIG. 12 is graphs showing evaluation results when simulating therelationship between η at which the secondary temperature coefficientβ=0, that is, the line occupation rate η indicating the tertiarytemperature characteristics, and the groove depth G. The quartz crystalsubstrate 30 has Euler angles of (0°, 123°, and Ψ). Herein, Ψ isappropriately selected as the angle at which the frequency-temperaturecharacteristics indicate the tendency of the tertiary curve, that is,the angle at which the secondary temperature coefficient β=0. Therelationship between the Euler angle and the groove depth G whenobtaining η at which β=0, under the same conditions as in FIG. 12, isshown in FIG. 34. Of FIG. 34, in the graph (FIG. 34(C)) in which theelectrode film thickness H=0.02λ, although the plots of Ψ≦42° are notshown, the η2 plot of the graph is Ψ=41.9° at G=0.03λ. The plots of therelationship between the groove depth G and line occupation rate η foreach electrode film thickness are obtained based on FIG. 15 to FIG. 20,which will be described in detail hereafter.

From the evaluation results shown in FIG. 12, it can be seen that forevery film thickness, as heretofore described, the fluctuation of η1 dueto the change in the groove depth G is small in comparison with that ofη2. For this reason, η1 has been extracted from the graphs showing therelationship between the groove depth G and line occupation rate η foreach film thickness in FIG. 12, and summarized by plotting points atwhich β≈0 in FIG. 13(A). In contrast, on evaluating an area in which,although not reaching β≈0, |β|≦0.01 is satisfied, it becomes clear thatη1 are concentrated in the polygon indicated by the solid line, as shownin FIG. 13(B).

The coordinates of points a to h in FIG. 13(B) are shown in Table 1below.

TABLE 1 Point G/λ η a 0.01 0.70 b 0.03 0.66 c 0.05 0.62 d 0.07 0.55 e0.07 0.60 f 0.05 0.65 g 0.03 0.70 h 0.01 0.75

FIG. 13(B) shows that, provided that it is inside the polygon surroundedby points a to h, |β|≦0.01 is guaranteed regardless of the size of theelectrode film thickness H, and good frequency-temperaturecharacteristics can be obtained. A range within which the goodfrequency-temperature characteristics can be obtained is a range whichsatisfies both Equations (11) and (12), and Equation (13), shown below.

[Expression 22]

η≦−2.5000×G/λ+0.7775

provided that

0.0100λ≦G≦0.0695λ  (1 1)

[Expression 23]

η≧2.0000×G/λ+0.7200

provided that

0.0100λ≦G≦0.0500λ  (1 2)

[Expression 24]

η≧−3.5898×G/λ+0.7995

provided that

0.0500λ<G≦0.0695λ  (1 3)

From Equations (11), (12), and (13), it can be said that it is possibleto specify the line occupation rates η in the range surrounded by thesolid line in FIG. 13(B) as a range satisfying both Equation (5) andEquation (6).

[Expression 25]

−2.0000×G/λ+0.7200≦η≦−2.5000×Gλ+0.7775

provided that

0.0100λ≦G≦0.05002  (5)

[Expression 26]

−3.5898×G/λ+0.7995≦η≦−2.5000×G/λ+0.7775

provided that

0.0500λ<G≦0.06952

Herein, in the case of allowing the secondary temperature coefficient βto within ±0.01 (ppm/° C.²), it is confirmed that by configuring in sucha way as to satisfy both Equation (3) and Equation (5) when0.0100λ≦G≦0.0500λ, and satisfy both Equation (3) and Equation (6) when0.0500λ≦G≦0.0695λ, the secondary temperature coefficient β comes within±0.01 (ppm/° C.²).

The values of the secondary temperature coefficient β for each electrodefilm thickness H at the points a to h are shown in Table 2 below. FromTable 2, it can be confirmed that |β|≦0.01 at all of the points.

TABLE 2 Electrode Film Thickness H Point 1% λ 1.5% λ 2% λ 2.5% λ 3% λ3.5% λ a −0.0099 −0.0070 −0.0030 0.0030 −0.0050 −0.0060 b 0.0040 0.00300.0000 0.0000 −0.0020 −0.0040 c 0.0070 −0.0040 0.0010 −0.0036 −0.0040−0.0057 d 0.0067 −0.0022 −0.0070 −0.0080 −0.0090 −0.0099 e −0.0039−0.0060 −0.0090 −0.0080 −0.0090 −0.0094 f −0.0023 −0.0070 −0.0050−0.0062 −0.0060 −0.0070 g −0.0070 −0.0060 −0.0090 −0.0070 −0.0070−0.0070 h −0.0099 −0.0030 −0.0091 −0.0080 −0.0080 −0.0080

Also, when the relationship between the groove depth G and lineoccupation rate η at each point at which β=0 for SAW resonators 10 inwhich the electrode film thickness H≈0, 0.1λ, 0.02λ, 0.03λ, or 0.035λ,based on Equations (11) to (13) and Equations (5) and (6) derivedtherefrom, is indicated by an approximate line, the result is as in FIG.14. The relationship between the groove depth G and line occupation rateη the quartz crystal substrate 30 on which no electrode film is providedis as shown in FIG. 9.

When changing the electrode film thickness H at 3.0% λ (0.030λ) or less,the frequency-temperature characteristics of β=0, that is, the tertiarycurve, can be obtained. At this time, a relational equation for G and ηwhen the frequency-temperature characteristics are good can be expressedby Equation (8).

[Expression 27]

η=−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−

135.99×(H/λ)²+5.817×(H/λ)+0.732

99.99×(G/λ)+(H/λ)  (8)

Herein, the units of G and H are λ.

It should be noted that Equation (8) is established when the electrodefilm thickness H is in the range of 0<H≦0.030λ.

The manufacturing variation of the electrical characteristics(particularly the resonance frequency) being greater the greater theelectrode film thickness, it is highly likely that the manufacturingvariation of the line occupation rate η is ±0.04 or less when theelectrode film thickness H is within the range of Equations (5) and (6),and that a manufacturing variation greater than ±0.04 occurs whenH>0.035λ. However, provided that the electrode film thickness H iswithin the range of Equations (5) and (6), and the variation of the lineoccupation rate η is ±0.04 or less, it is possible to realize a SAWdevice with a low secondary temperature coefficient β. That is, whentaking into consideration the manufacturing variation of the lineoccupation rate, and keeping the secondary temperature coefficient βwithin ±0.01 ppm/° C.², a line occupation rate η up to the range ofEquation (9), wherein a tolerance of ±0.04 is added to Equation (8), isallowable.

[Expression 28]

η=−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−

135.99×(H/λ)²+5.817×(H/λ)+0.732−

99.99×(G/λ)×(H/λ)±0.04  (9)

FIGS. 15 to 20 show graphs of the relationship between the lineoccupation rate η and secondary temperature coefficient β when thegroove depth G is changed, in cases in which the electrode filmthickness is 0.01λ (1% λ), 0.015λ (1.5% λ), 0.02λ (2% λ), 0.025λ (2.5%λ), 0.03λ (3% λ), and 0.035λ (3.5% λ), respectively.

Also, FIGS. 21 to 26 show graphs of the relationship between the lineoccupation rate η and frequency fluctuation amount ΔF in the SAWresonator 10 corresponding to each of the FIGS. 15 to 20. The quartzcrystal substrates used are all ones with Euler angles (0°, 123°, andΨ), and with regard to Ψ, an angle at which ΔF is smallest isappropriately selected.

Herein, FIG. 15 are diagrams showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when theelectrode film thickness H is 0.01λ, and FIG. 21 are diagrams showingthe relationship between the line occupation rate η and frequencyfluctuation amount ΔF when the electrode film thickness H is 0.01λ.

Also, FIG. 16 are diagrams showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when theelectrode film thickness H is 0.015λ, and FIG. 22 are diagrams showingthe relationship between the line occupation rate η and frequencyfluctuation amount ΔF when the electrode film thickness H is 0.015λ.

Also, FIG. 17 are diagrams showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when theelectrode film thickness H is 0.02λ, and FIG. 23 are diagrams showingthe relationship between the line occupation rate η and frequencyfluctuation amount ΔF when the electrode film thickness H is 0.02λ.

Also, FIG. 18 are diagrams showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when theelectrode film thickness H is 0.025λ, and FIG. 24 are diagrams showingthe relationship between the line occupation rate η and frequencyfluctuation amount ΔF when the electrode film thickness H is 0.025λ.

Also, FIG. 19 are diagrams showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when theelectrode film thickness H is 0.03λ, and FIG. 25 are diagrams showingthe relationship between the line occupation rate η and frequencyfluctuation amount ΔF when the electrode film thickness H is 0.03λ.

Also, FIG. 20 are diagrams showing the relationship between the lineoccupation rate η and secondary temperature coefficient β when theelectrode film thickness H is 0.035λ, and FIG. 26 are diagrams showingthe relationship between the line occupation rate η and frequencyfluctuation amount ΔF when the electrode film thickness H is 0.035λ.

Although there are slight differences in all of the graphs in thesediagrams (FIGS. 15 to 26), regarding the change tendencies thereof, itis seen that they are similar to those in FIGS. 8 and 10, which aregraphs showing the relationship between the line occupation rate η andsecondary temperature coefficient β, and line occupation rate η andfrequency fluctuation amount ΔF, in the quartz crystal substrate 30only.

That is, it can be said that an advantage according to the invention isthat it can be accomplished even when propagating a surface acousticwave on an individual quartz crystal substrate 30 from which theelectrode film is omitted.

For each of the two points η1 and η2 at which the secondary temperaturecoefficient β becomes zero, a simulation is performed for each of therange of η1 and η2 when the range of β is expanded to |β|≦0.01, and thecase in which the range of the electrode film thickness H is fixed, andthe groove depth G is changed. Of η1 and η2, the larger η at which|β|≦0.01 is taken to be η1, and the smaller η at which |β|≦0.01 is η2.The quartz crystal substrates used are all ones with Euler angles (0°,123°, and Ψ), and with regard to Ψ, an angle at which ΔF is smallest isappropriately selected.

FIG. 27(A) is a graph showing the relationship between η1 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.000λ<H≦0.005λ, and Table 3 is atable showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 27(A), and the value of β at themeasurement points.

TABLE 3 Point G/λ η β a 0.0100 0.7100 −0.0098 b 0.0200 0.7100 −0.0099 c0.0300 0.7100 −0.0095 d 0.0400 0.7100 −0.0100 e 0.0500 0.7100 −0.0100 f0.0600 0.7100 −0.0098 g 0.0700 0.7100 −0.0099 h 0.0800 0.7100 −0.0097 i0.0900 0.7100 −0.0100 j 0.0900 0.4200 0.0073 k 0.0800 0.5700 0.0086 l0.0700 0.5900 0.0093 m 0.0600 0.6150 0.0077 n 0.0500 0.6300 0.0054 o0.0400 0.6350 0.0097 p 0.0300 0.6500 0.0097 q 0.0200 0.6700 0.0074 r0.0100 0.7100 0.0091

From FIG. 27(A) and Table 3, it can be seen that when the electrode filmthickness H at η1 is within the heretofore described range, and when thegroove depth G is in the range 0.01λ≦G≦0.09λ, β satisfies the heretoforedescribed requirement in the area surrounded by the measurement points ato r.

FIG. 27(B) is a graph showing the relationship between η2 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.000λ<H≦0.005λ, and Table 4 is atable showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 27(B), and the value of β at themeasurement points.

TABLE 4 Point G/λ η β a 0.0300 0.5900 0.0097 b 0.0400 0.5800 0.0097 c0.0500 0.5500 0.0054 d 0.0600 0.5200 0.0077 e 0.0700 0.4800 0.0093 f0.0800 0.4500 0.0086 g 0.0900 0.4000 0.0073 h 0.0900 0.1800 0.0056 i0.0800 0.3400 0.0093 j 0.0700 0.4100 0.0078 k 0.0600 0.4600 0.0094 l0.0500 0.4900 0.0085 m 0.0400 0.5200 0.0099 n 0.0300 0.5500 0.0098

From FIG. 27(B) and Table 4, it can be seen that when the electrode filmthickness H at η2 is within the heretofore described range, and when thegroove depth G is in the range of 0.03λ≦G≦0.09λ, β satisfies theheretofore described requirement in the area surrounded by themeasurement points a to n.

FIG. 28(A) is a graph showing the relationship between η1 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.005λ<H≦0.010λ, and Table 5 is atable showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 28(A), and the value of β at themeasurement points.

TABLE 5 Point G/λ η β a 0.0100 0.7700 −0.0099 b 0.0200 0.7400 −0.0100 c0.0300 0.7150 −0.0100 d 0.0400 0.7300 −0.0098 e 0.0500 0.7400 −0.0100 f0.0600 0.7300 −0.0098 g 0.0700 0.7300 −0.0100 h 0.0800 0.7300 −0.0100 i0.0800 0.5000 0.0086 j 0.0700 0.5700 0.0100 k 0.0600 0.6100 0.0095 l0.0500 0.6300 0.0100 m 0.0400 0.6350 0.0097 n 0.0300 0.6550 0.0070 o0.0200 0.6800 0.0100 p 0.0100 0.7600 0.0016

From FIG. 28(A) and Table 5, it can be seen that when the electrode filmthickness H at η1 is within the heretofore described range, and when thegroove depth G is in the range of 0.01λ≦G≦0.08λ, β satisfies theheretofore described requirement in the area surrounded by themeasurement points a to p.

FIG. 28(B) is a graph showing the relationship between η2 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.005λ<H≦0.010λ, and Table 6 is atable showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 28(B), and the value of β at themeasurement points.

TABLE 6 Point G/λ η β a 0.0200 0.6500 0.0090 b 0.0300 0.6100 0.0098 c0.0400 0.5700 0.0097 d 0.0500 0.5500 0.0040 e 0.0600 0.5200 0.0066 f0.0700 0.4700 0.0070 g 0.0700 0.3700 −0.0094 h 0.0600 0.4400 −0.0096 i0.0500 0.4800 −0.0096 j 0.0400 0.5200 −0.0095 k 0.0300 0.5500 −0.0099 l0.0200 0.5900 −0.0100

From FIG. 28(B) and Table 6, it can be seen that when the electrode filmthickness H at η2 is within the heretofore described range, and when thegroove depth G is in the range of 0.02λ≦G≦0.07λ, β satisfies theheretofore described requirement in the area surrounded by themeasurement points a to 1.

FIG. 29(A) is a graph showing the relationship between η1 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.010λ<H≦0.015λ, and Table 7 is atable showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 29(A), and the value of β at themeasurement points.

TABLE 7 Point G/λ η β a 0.0100 0.770 −0.0099 b 0.0200 0.760 −0.0099 c0.0300 0.760 −0.0099 d 0.0400 0.750 −0.0099 e 0.0500 0.750 −0.0099 f0.0600 0.750 −0.0099 g 0.0700 0.740 −0.0099 h 0.0800 0.740 −0.0098 i0.0800 0.340 0.0088 j 0.0700 0.545 0.0088 k 0.0600 0.590 0.0099 l 0.05000.620 0.0090 m 0.0400 0.645 0.0060 n 0.0300 0.670 0.0030 o 0.0200 0.7050.0076 p 0.0100 0.760 0.0010

From FIG. 29(A) and Table 7, it can be seen that when the electrode filmthickness H at η1 is within the heretofore described range, and when thegroove depth G is in the range of 0.01λ≦G≦0.08μ, β satisfies theheretofore described requirement in the area surrounded by themeasurement points a to p.

FIG. 29(B) is a graph showing the relationship between η2 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.010λ<H≦0.015λ, and Table 8 is atable showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 29(B), and the value of β at themeasurement points.

TABLE 8 Point G/λ η β a 0.0100 0.740 0.0099 b 0.0200 0.650 0.0090 c0.0300 0.610 0.0090 d 0.0400 0.570 0.0080 e 0.0500 0.540 0.0060 f 0.06000.480 0.0060 g 0.0700 0.430 0.0099 h 0.0700 0.3500 −0.0099 i 0.06000.4200 −0.0090 j 0.0500 0.4700 −0.0090 k 0.0400 0.5100 −0.0090 l 0.03000.5500 −0.0090 m 0.0200 0.6100 −0.0099 n 0.0100 0.7000 −0.0099

From FIG. 29(B) and Table 8, it can be seen that when the electrode filmthickness H at η2 is within the heretofore described range, and when thegroove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfies theheretofore described requirement in the area surrounded by themeasurement points a to n.

FIG. 30(A) is a graph showing the relationship between η1 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.015λ<H≦0.020λ, and Table 9 is atable showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 30(A), and the value of β at themeasurement points.

TABLE 9 Point G/λ η β a 0.010 0.770 −0.0100 b 0.020 0.770 −0.0100 c0.030 0.760 −0.0100 d 0.040 0.760 −0.0100 e 0.050 0.760 −0.0100 f 0.0600.750 −0.0100 g 0.070 0.750 −0.0100 h 0.070 0.510 0.0100 i 0.060 0.5700.0099 j 0.050 0.620 0.0097 k 0.040 0.640 0.0096 l 0.030 0.660 0.0080 m0.020 0.675 0.0076 n 0.010 0.700 0.0010

From FIG. 30(A) and Table 9, it can be seen that when the electrode filmthickness H at η1 is within the heretofore described range, and when thegroove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfies theheretofore described requirement in the area surrounded by themeasurement points a to n.

FIG. 30(B) is a graph showing the relationship between η2 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.015λ<H≦0.020λ, and Table 10 isa table showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 30(B), and the value of β at themeasurement points.

TABLE 10 Point G/λ η β a 0.010 0.690 0.0010 b 0.020 0.640 0.0090 c 0.0300.590 0.0090 d 0.040 0.550 0.0080 e 0.050 0.510 0.0080 f 0.060 0.4700.0090 g 0.070 0.415 0.0100 h 0.070 0.280 −0.0100 i 0.060 0.380 −0.0090j 0.050 0.470 −0.0090 k 0.040 0.510 −0.0090 l 0.030 0.550 −0.0090 m0.020 0.610 −0.0100 n 0.010 0.680 −0.0100

From FIG. 30(B) and Table 10, it can be seen that when the electrodefilm thickness H at η2 is within the heretofore described range, andwhen the groove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfiesthe heretofore described requirement in the area surrounded by themeasurement points a to n.

FIG. 31(A) is a graph showing the relationship between η1 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.020λ<H≦0.025λ, and Table 11 isa table showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 31(A), and the value of β at themeasurement points.

TABLE 11 Point G/λ η β a 0.010 0.770 −0.0100 b 0.020 0.770 −0.0100 c0.030 0.760 −0.0100 d 0.040 0.760 −0.0100 e 0.050 0.760 −0.0096 f 0.0600.760 −0.0100 g 0.070 0.760 −0.0100 h 0.070 0.550 0.0100 i 0.060 0.5450.0090 j 0.050 0.590 0.0097 k 0.040 0.620 0.0100 l 0.030 0.645 0.0100 m0.020 0.680 0.0070 n 0.010 0.700 0.0030

From FIG. 31(A) and Table 11, it can be seen that when the electrodefilm thickness H at η1 is within the heretofore described range, andwhen the groove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfiesthe heretofore described requirement in the area surrounded by themeasurement points a to n.

FIG. 31(B) is a graph showing the relationship between η2 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.020λ<H≦0.025λ, and Table 12 isa table showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 31(B), and the value of β at themeasurement points.

TABLE 12 Point G/λ η β a 0.010 0.690 0.0030 b 0.020 0.640 0.0090 c 0.0300.590 0.0090 d 0.040 0.550 0.0090 e 0.050 0.510 0.0080 f 0.060 0.4200.0090 g 0.070 0.415 0.0080 h 0.070 0.340 −0.0098 i 0.060 0.340 −0.0100j 0.050 0.420 −0.0100 k 0.040 0.470 −0.0100 l 0.030 0.520 −0.0093 m0.020 0.580 −0.0100 n 0.010 0.650 −0.0090

From FIG. 31(B) and Table 12, it can be seen that when the electrodefilm thickness H at η2 is within the heretofore described range, andwhen the groove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfiesthe heretofore described requirement in the area surrounded by themeasurement points a to n.

FIG. 32(A) is a graph showing the relationship between η1 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.025λ<H≦0.030λ, and Table 13 isa table showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 32(A), and the value of β at themeasurement points.

TABLE 13 Point G/λ η β a 0.010 0.770 −0.0098 b 0.020 0.770 −0.0100 c0.030 0.770 −0.0100 d 0.040 0.760 −0.0100 e 0.050 0.760 −0.0099 f 0.0600.760 −0.0100 g 0.070 0.760 −0.0100 h 0.070 0.550 0.0080 i 0.060 0.5050.0087 j 0.050 0.590 0.0090 k 0.040 0.620 0.0100 l 0.030 0.645 0.0100 m0.020 0.680 0.0083 n 0.010 0.700 0.0052

From FIG. 32(A) and Table 13, it can be seen that when the electrodefilm thickness H at η1 is within the heretofore described range, andwhen the groove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfiesthe heretofore described requirement in the area surrounded by themeasurement points a to n.

FIG. 32(B) is a graph showing the relationship between η2 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.025λ<H≦0.030λ, and Table 14 isa table showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 32(B), and the value of β at themeasurement points.

TABLE 14 Point G/λ η β a 0.010 0.670 0.0052 b 0.020 0.605 0.0081 c 0.0300.560 0.0092 d 0.040 0.520 0.0099 e 0.050 0.470 0.0086 f 0.060 0.3950.0070 g 0.070 0.500 0.0080 h 0.070 0.490 −0.0100 i 0.060 0.270 −0.0100j 0.050 0.410 −0.0100 k 0.040 0.470 −0.0100 l 0.030 0.520 −0.0093 m0.020 0.580 −0.0099 n 0.010 0.620 −0.0090

From FIG. 32(B) and Table 14, it can be seen that when the electrodefilm thickness H at η2 is within the heretofore described range, andwhen the groove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfiesthe heretofore described requirement in the area surrounded by themeasurement points a to n.

FIG. 33(A) is a graph showing the relationship between η1 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.030λ<H≦0.035λ, and Table 15 isa table showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 33(A), and the value of β at themeasurement points.

TABLE 15 Point G/λ η β a 0.010 0.770 −0.0100 b 0.020 0.770 −0.0098 c0.030 0.770 −0.0100 d 0.040 0.760 −0.0100 e 0.050 0.760 −0.0100 f 0.0600.760 −0.0100 g 0.070 0.760 −0.0100 h 0.070 0.550 0.0090 i 0.060 0.5000.0087 j 0.050 0.545 0.0090 k 0.040 0.590 0.0091 l 0.030 0.625 0.0080 m0.020 0.650 0.0083 n 0.010 0.680 0.0093

From FIG. 33(A) and Table 15, it can be seen that when the electrodefilm thickness H at η1 is within the heretofore described range, andwhen the groove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfiesthe heretofore described requirement in the area surrounded by themeasurement points a to n.

FIG. 33(B) is a graph showing the relationship between η2 whichsatisfies the heretofore described range of β and the groove depth G,when the electrode film thickness H is 0.030λ<H≦0.035λ, and Table 16 isa table showing the coordinates (G/λ, η) of principal measurement pointsfor fixing the range shown in FIG. 33(B), and the value of β at themeasurement points.

TABLE 16 Point G/λ η β a 0.010 0.655 0.0080 b 0.020 0.590 0.0081 c 0.0300.540 0.0092 d 0.040 0.495 0.0099 e 0.050 0.435 0.0090 f 0.060 0.3950.0061 g 0.070 0.500 0.0090 h 0.070 0.550 −0.0100 i 0.060 0.380 −0.0090j 0.050 0.330 −0.0100 k 0.040 0.410 −0.0095 l 0.030 0.470 −0.0099 m0.020 0.520 −0.0100 n 0.010 0.590 −0.0100

From FIG. 33(B) and Table 16, it can be seen that when the electrodefilm thickness H at η2 is within the heretofore described range, andwhen the groove depth G is in the range of 0.01λ≦G≦0.07λ, β satisfiesthe heretofore described requirement in the area surrounded by themeasurement points a to n.

The relationship between Ψ and the groove depth G obtained from η1 inthe graphs shown in FIG. 34 is summarized in FIG. 35. The reason forselecting η1 is as heretofore described. As shown in FIG. 35, there ishardly any change in the angle Ψ, even though the thickness of theelectrode film changes, and it is seen that the optimum angle Ψ changesin accordance with the fluctuation of the groove depth G. It can be saidthat this too is proof that a high proportion of the change in thesecondary temperature coefficient β is due to the form of the quartzcrystal substrate 30.

In the same way as heretofore described, the relationships between thegroove depth G and Ψ when the secondary temperature coefficient β=−0.01(ppm/° C.²), and Ψ when β=+0.01 (ppm/° C.²), are obtained, andsummarized in FIGS. 36 and 37. By obtaining from these graphs (FIGS. 35to 37) the angles Ψ which it is possible to achieve −0.01≦β≦+0.01, it ispossible to fix the preferable Ψ angle range under the heretoforedescribed conditions at 43°<Ψ<45°, and it is possible to more preferablyfix the range at 43.2°≦Ψ≦44.2.

A simulation is carried out for the range of Ψ which satisfies therequirement |β|≦0.01 when changing the groove depth G, in the case ofchanging the electrode film thickness H. The results of the simulationare shown in FIGS. 38 to 44. The quartz crystal substrates used are allones with Euler angles (0°, 123°, and Ψ), and with regard to Ψ, an angleat which ΔF is smallest is appropriately selected.

FIG. 38(A) is a graph showing the range of Ψ which satisfies therequirement |β|≦0.01 when the range of the electrode film thickness H is0<H≦0.005λ. Herein, the range sandwiched by the straight line connectingthe plots indicating the maximum value of Ψ and the broken lineconnecting the plots indicating the minimum value of Ψ is the rangewhich satisfies the heretofore described condition.

With the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line shown in FIG.38(A) as a polygon, it can be shown as in FIG. 38(B), and it can be saidthat β satisfies the heretofore described condition in a rangecorresponding to the interior of the polygon shown by the solid lines inFIG. 38(B). When expressing the range of the polygon shown in FIG. 38(B)in an approximate equation, it can be expressed by Equations (14) and(15).

[Expression 29]

Ψ≦3.0×G/λ+43.92

provided that

0.0100λ≦G≦0.0695λ  (1 4)

[Expression 30]

Ψ≧−48.0×G/λ44.35

provided that

0.0100λ≦G≦0.0695λ  (1 5)

FIG. 39(A) is a graph showing the range of which satisfies therequirement |β|≦0.01 when the range of the electrode film thickness H is0.005λ<H≦0.010λ. Herein, the range sandwiched by the straight lineconnecting the plots indicating the maximum value of Ψ and the brokenline connecting the plots indicating the minimum value of Ψ is the rangewhich satisfies the heretofore described condition.

With the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line in FIG. 39(A)as a polygon, it can be shown as in FIG. 29(B), and it can be said thatβ satisfies the heretofore described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 39(B).When expressing the range of the polygon shown in FIG. 39(B) in anapproximate equation, it can be expressed by Equations (16) and (17).

[Expression 31]

Ψ≦8.0×G/λ+43.60

provided that

0.0100λ≦G≦0.0695λ  (1 6)

[Expression 32]

Ψ≧−48.0×G/λ+44.00

provided that

0.0100λ≦G≦0.0695λ  (1 7)

FIG. 40(A) is a graph showing the range of Ψ which satisfies therequirement |β|≦0.01 when the range of the electrode film thickness H is0.010λ<H≦0.015λ. Herein, the range sandwiched by the straight lineconnecting the plots indicating the maximum value of Ψ and the brokenline connecting the plots indicating the minimum value of Ψ is the rangewhich satisfies the heretofore described condition.

With the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line in FIG. 40(A)as a polygon, it can be shown as in FIG. 40(B), and it can be said thatβ satisfies the heretofore described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 40(B).When expressing the range of the polygon shown in FIG. 40(B) in anapproximate equation, it can be expressed by Equations (18) and (19).

[Expression 33]

Ψ≦10.0×G/λ+43.40

provided that

0.0100λ≦G≦0.0695λ  (1 8)

[Expression 34]

Ψ≧−44.0×G/λ+43.80

provided that

0.0100λ≦G≦0.0695λ(1 9)

FIG. 41(A) is a graph showing the range of Ψ which satisfies therequirement |β|≦0.01 when the range of the electrode film thickness H is0.015<H≦0.020λ. Herein, the range sandwiched by the straight lineconnecting the plots indicating the maximum value of Ψ and the brokenline connecting the plots indicating the minimum value of Ψ is the rangewhich satisfies the heretofore described condition.

With the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line in FIG. 41(A)as a polygon, it can be shown as in FIG. 41(B), and it can be said thatβ satisfies the heretofore described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 41(B).When expressing the range of the polygon shown in FIG. 41(B) in anapproximate equation, it can be expressed by Equations (20) and (21).

[Expression 35]

Ψ≦12.0×G/λ+43.31

provided that

0.0100λ≦G≦0.0695λ  (2 0)

[Expression 36]

Ψ≧−30.0×G/λ+44.40

provided that

0.0100λ≦G≦0.0695λ  (2 1)

FIG. 42(A) is a graph showing the range of Ψ which satisfies therequirement |β|≦0.01 when the range of the electrode film thickness H is0.020λ<H≦0.025λ. Herein, the range sandwiched by the straight lineconnecting the plots indicating the maximum value of Ψ and the brokenline connecting the plots indicating the minimum value of Ψ is the rangewhich satisfies the heretofore described condition.

With the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line in FIG. 42(A)as a polygon, it can be shown as in FIG. 42(B) and it can be said that βsatisfies the heretofore described condition in a range corresponding tothe interior of the polygon shown by the solid lines in FIG. 42(B). Whenexpressing the range of the polygon shown in FIG. 42(B) in anapproximate equation, it can be expressed by Equations (22) to (24).

[Expression 37]

Ψ≦14.0×G/λ+43.16

provided that

0.0100λ≦G≦0.0695λ  (2 2)

[Expression 38]

Ψ≧−45.0×G/λ+43.35

provided that

0.0100λ≦G≦0.0600λ  (2 3)

[Expression 39]

Ψ≧367.368×G/λ+18.608

provided that

0.0600λ≦G≦0.0695λ  (2 4)

FIG. 43(A) is a graph showing the range of Ψ which satisfies therequirement |β|≦0.01 when the range of the electrode film thickness H is0.025λ<H≦0.030λ. Herein, the range sandwiched by the straight lineconnecting the plots indicating the maximum value of Ψ and the brokenline connecting the plots indicating the minimum value of Ψis the rangewhich satisfies the heretofore described condition.

With the groove depth G in the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line in FIG. 43(A)as a polygon, it can be shown as in FIG. 43(B), and it can be said thatβ satisfies the heretofore described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 43(B).When expressing the range of the polygon shown in FIG. 43(B) in anapproximate equation, it can be expressed by Equations (25) to (27).

[Expression 40]

Ψ≧12.0×G/λ+43.25

provided that

0.0100λ≦G≦0.0695λ  (2 5)

[Expression 41]

Ψ≧−50.0×G/λ+43.32

provided that

0.0100λ≦G≦0.0500λ  (2 6)

[Expression 42]

Ψ≧167.692×G/λ+32.435

provided that

0.0500λ≦G≦0.0695λ  (2 7)

FIG. 44(A) is a graph showing the range of which satisfies therequirement |β|≦0.01 when the range of the electrode film thickness H is0.030λ<H≦0.035λ. Herein, the range sandwiched by the straight lineconnecting the plots indicating the maximum value of Ψ and the brokenline connecting the plots indicating the minimum value of Ψ is the rangewhich satisfies the heretofore described condition.

With the groove depth Gin the range of 0.01λ≦G≦0.0695λ, whenapproximating the range of the solid line and broken line in FIG. 44(A)as a polygon, it can be shown as in FIG. 44(B), and it can be said thatβ satisfies the heretofore described condition in a range correspondingto the interior of the polygon shown by the solid lines in FIG. 44(B).When expressing the range of the polygon shown in FIG. 44(B) in anapproximate equation, it can be expressed by Equations (28) to (30).

[Expression 43]

Ψ≧12.0×G/λ+43.35

provided that

0.0100λ≦G≦0.0695λ(2 8)

[Expression 44]

Ψ≧−45.0×G/λ+42.80

provided that

0.0100λ≦G≦0.0500λ  (2 9)

[Expression 45]

Ψ≧186.667×G/λ+31.217

provided that

0.0500λ≦G≦0.0695λ  (3 0)

Next, the change in the secondary temperature coefficient β when theangle θ is altered, that is, the relationship between θ and thesecondary temperature coefficient β, is shown in FIG. 45. Herein, theSAW device used in the simulation is a quartz crystal substrate with cutangles and SAW propagation direction of (0, θ, and Ψ) in Euler anglerepresentation, with a groove depth G of 0.04λ, and the electrode filmthickness H is 0.02λ. With regard to Ψ, a value at which the absolutevalue of the secondary temperature coefficient β is smallest in theheretofore described angle range is appropriately selected, based on theθ setting angle. Also, with regard to η, it is 0.6383, in accordancewith Equation (8).

Under these kinds of condition, from FIG. 45, which shows therelationship between θ and the secondary temperature coefficient β, itcan be seen that, provided that θ is within the range of 117° or more to142° or less, the absolute values of the secondary temperaturecoefficient β are within a range of 0.01 (ppm/° C.²). Therefore, it canbe said that, with the heretofore described kinds of setting value, byfixing θ in the range of 117°≦θ≦142°, it is possible to configure a SAWresonator 10 which has good frequency-temperature characteristics.

Tables 17 to 19 are shown as simulation data proving the relationshipbetween θ and the secondary temperature coefficient β.

TABLE 17 H/λ G/λ θ β % % ° ppm/° C.² 0.01 4.00 117 −0.009 0.01 4.00 1420.005 3.50 4.00 117 −0.009 3.50 4.00 142 −0.008Table 17 being a table showing the relationship between θ and thesecondary temperature coefficient β when the electrode film thickness His changed, it shows the values of the secondary temperature coefficientβ at the critical values (117° and 142°) of θ when the electrode filmthickness H is 0.01% λ, and when the electrode film thickness H is 3.50%λ. The groove depths G in the simulation are all 4% λ. From Table 17, itcan be seen that, in the range of 117°≦θ≦142°, even though the thicknessof the electrode film thickness H is changed (0≈0.01% λ and 3.5% λstipulated as critical values of the electrode film thickness), |β|≦0.01is satisfied regardless of the thickness.

TABLE 18 H/λ G/λ θ β % % ° ppm/° C.² 2.00 1.00 117 −0.009 2.00 1.00 142−0.008 2.00 6.95 117 −0.009 2.00 6.95 142 −0.009

Table 18 being a table showing the relationship between θ and thesecondary temperature coefficient β when the groove depth G is changed,it shows the values of the secondary temperature coefficient β at thecritical values (117° and 142°) of θ when the groove depth G is 1.00% λand 6.95% λ. The electrode film thicknesses H in the simulation are all2.00% λ. From Table 18, it can be seen that, in the range of117°≦θ≦142°, even though the groove depth G is changed (1.00% λ, and6.95% λ stipulated as critical values of the groove depth G), |β|≦0.01is satisfied regardless of the depth.

TABLE 19 H/λ G/λ θ β % % η ° ppm/° C.² 2.00 4.00 0.62 117 −0.010 2.004.00 0.62 142 −0.003 2.00 4.00 0.76 117 −0.009 2.00 4.00 0.76 142 −0.009

Table 19 being a table showing the relationship between θ and thesecondary temperature coefficient β when the line occupation rate η ischanged, it shows the values of the secondary temperature coefficient βat the critical values (117° and 142°) of θ when the line occupationrate η is 0.62 and 0.76. The electrode film thicknesses H in thesimulation are all 2.00% λ, and the groove depths G are all 4.00% λ.From Table 19, it can be seen that, in the range of 117°≦θ≦142°, eventhough the line occupation rate η is changed (η=0.62 and 0.76 are theminimum value and maximum value of η when the groove depth is 40% λ inFIG. 31(A), which shows the relationship between the line occupationrate η (η1) and groove depth G with the electrode film thickness H inthe range of 0.020λ to 0.025), |β|≦0.01 is satisfied regardless of thevalue.

FIG. 46 is a graph showing the relationship between the angle φ and thesecondary temperature coefficient β in the case of using a quartzcrystal substrate 30 of (φ, 123°, and 43.77°) in Euler anglerepresentation, with a groove depth G of 0.04λ, an electrode filmthickness H of 0.02λ, and a line occupation rate η of 0.65.

From FIG. 46, it can be seen that when φ is −2° and +2°, the secondarytemperature coefficient β is lower than −0.01 in each case, but providedthat φ is in the range of −1.5° to +1.5°, the absolute values of thesecondary temperature coefficient β are consistently within a range of0.01. Therefore, with the heretofore described kinds of setting value,by fixing φ in the range of −1.5°≦φ≦+1.5°, or preferably −1°≦φ≦+1°, itis possible to configure a SAW resonator 10 which has goodfrequency-temperature characteristics.

In the above description, the range of optimum values of each of φ, θ,and Ψ r is derived for the relationship with the groove depth G underconstant conditions. In contrast to this, an extremely preferablerelationship between θ and Ψ, wherein the frequency fluctuation amountat −40° C. to +85° C. is smallest, is shown in FIG. 47, and anapproximate equation is obtained. According to FIG. 47, the angle Ψchanges along with an increase of the angle θ, increasing in such a wayas to describe a tertiary curve. In the example of FIG. 47, Ψ is 42.79°when θ=117°, and Ψ is 49.57° when θ=142°. When showing these plots as anapproximate curve, they become the curve shown by the broken line inFIG. 47, and can be expressed as an approximate equation by Equation(31)

[Expression 46]

Ψ=1.19024×10³×θ³−4.48775×10⁻¹×θ2+5.64362×10¹×θ−2.32327×10 ³±1.0  (3 1)

Because of this, it is possible to fix Ψ by θ being fixed, and it ispossible to make the range of Ψ 42.79°≦Ψ≦49.57° when the range of θ is117°≦θ≦142°. The groove depth G and electrode film thickness H in thesimulation are G=0.04λ, and H=0.02λ.

Due to the heretofore described kinds of reason, by configuring the SAWresonator 10 in accordance with the various conditions fixed in theembodiment, it is possible to obtain a SAW resonator which can realizegood frequency-temperature characteristics fulfilling the target values.

Also, with the SAW resonator 10 according to the embodiment, theimprovement in the frequency-temperature characteristics is sought byhaving the film thickness H of the electrode film in the range of0<H≦0.035λ, as shown in Equation (7) and FIGS. 15 to 26. This, differingfrom the heretofore known way of seeking improvement in thefrequency-temperature characteristics by making the film thickness Hextremely large, realizes the improvement in the frequency-temperaturecharacteristics while still maintaining environmental resistancecharacteristics. The relationship between the electrode film thickness(Al electrode film thickness) and frequency fluctuation in a heat cycletest is shown in FIG. 54. The results of the heat cycle test shown inFIG. 54 are taken after a cycle in which the SAW resonator is exposed toan ambience of −55° C. for 30 minutes, then exposed for 30 minutes withthe ambient temperature raised to +125° C., is repeated eight times.From FIG. 54, it can be seen that the frequency fluctuation (Ffluctuation) in the range of the electrode film thickness H of the SAWresonator 10 according to the embodiment is ⅓ or less in comparison withthat of a case in which the electrode film thickness H is 0.06λ, and inwhich no inter-electrode finger groove is provided. For every plot inFIG. 54, H+G=0.06λ.

Also, on carrying out a high temperature storage test on a SAW resonatormanufactured under the same conditions as those in FIG. 54, exposing itto an ambience of 125° C. for 1,000 hours, it is confirmed that thefrequency fluctuation amount of the SAW resonator according to theembodiment before and after testing (the four conditions H=0.03λ andG=0.03λ, H=0.02λ and G=0.04λ, H=0.015λ and G=0.045λ, and H=0.01λ andG=0.05λ) is ⅓ or less in comparison with that of the heretofore knownSAW resonator (H=0.06λ and G=0).

With the SAW resonator 10 manufactured under the heretofore describedkinds of condition, conditions wherein H+G=0.067λ (aluminum filmthickness 2,000 Å, groove depth 4,700 Å), the IDT line occupation rateηi=0.6, the reflector line occupation rate ηr=0.8, the Euler angles are(0°, 123°, and 43.5°), the IDT pair number is 120, the intersectionwidth is 40λ (λ=10 μm), the number of reflectors (per side) is 72 (36pairs), and the electrode fingers have no angle of tilt (the electrodefinger array direction and SAW phase velocity direction correspond),frequency-temperature characteristics like those shown in FIG. 48 areexhibited.

In FIG. 48, the frequency-temperature characteristics are plotted forn=4 test specimens. According to FIG. 48, it can be seen that thefrequency fluctuation amount ΔF in the operating temperature range ofthe test specimens is kept to 20 ppm or less.

In the embodiment, a description has been given of the effect on thefrequency-temperature characteristics of the groove depth G, electrodefilm thickness H, and the like. However, the combined depth of thegroove depth G and electrode film thickness H (the level difference)also has an effect on static characteristics such as the equivalentcircuit constant and CI value, and on the Q value. For example, FIG. 49is a graph showing the relationship between the level difference and CIvalue when changing the level difference from 0.062λ to 0.071λ.According to FIG. 49, it can be seen that the CI value converges whenthe level difference is 0.067λ, and does not improve (does not becomelower) even when increasing the level difference further.

The frequency, equivalent circuit constant, and static characteristicsof the SAW resonator 10 exhibiting frequency-temperature characteristicslike those shown in FIG. 48 are collected in FIG. 50. Herein, Frepresents the frequency, Q the Q value, γ the capacity ratio, CI the CI(crystal impedance) value, and M the figure of merit (figure of merit).

Also, FIG. 52 shows a graph for comparing the relationship between thelevel difference and Q value for the heretofore known SAW resonator andthe SAW resonator 10 according to the embodiment. In FIG. 52, the graphshown with the thick line indicating the characteristics of the SAWresonator 10 according to the embodiment, grooves are provided betweenthe electrode fingers, and the stop band upper end mode resonance isused. The graph shown with the thin line indicating the characteristicsof the heretofore known SAW resonator, the stop band upper end moderesonance is used with no groove being provided between the electrodefingers. As is clear from FIG. 52, when providing the grooves betweenthe electrode fingers, and using the stop band upper end mode resonance,a higher Q value is obtained in the area in which the level difference(G+H) is 0.0407λ (4.07% λ) or more than in the case of using the stopband lower end mode resonance with no groove being provided between theelectrode fingers.

The basic data of the SAW resonators according to the simulation are asfollows.

Basic data of SAW resonator 10 according to the embodiment

H: 0.02λ

G: changesIDT line occupation rate ηi: 0.6Reflector line occupation rate ηr: 0.8Euler angles (0°, 123°, and 43.5°)Pair number: 120Intersection width: 40λ (λ=10 μm)Reflector number (per side): 60No electrode finger tilt angle

Basic data of heretofore known SAW resonator

H: changesG: zeroIDT line occupation rate ηi: 0.4Reflector line occupation rate ηr: 0.3Euler angles (0°, 123°, and 43.5°)Pair number: 120Intersection width: 40λ (λ=10 μm)Reflector number (per side): 60No electrode finger tilt angle

When referring to FIGS. 50 and 52 in order to compare thecharacteristics of the SAW resonators, it can be understood how muchhigher is the Q of the SAW resonator 10 according to the embodiment. Itcan be supposed that this kind of high Q is due to an energy confinementeffect, for the following reasons.

In order to efficiently confine the energy of a surface acoustic waveexcited in the stop band upper end mode, a stop band upper end frequencyft2 of the IDT 12 should be set between a stop band lower end frequencyfr1 of the reflectors 20 and a stop band upper end frequency fr2 of thereflectors 20, as in FIG. 53. That is, it should be set in such a way asto satisfy the relationship of

[Expression 47]

fr1<ft2<fr2  (3 2)

Because of this, a reflection coefficient Γ of the reflectors 20 at thestop band upper end frequency ft2 of the IDT 12 increases, and the stopband upper end mode SAW excited with the IDT 12 is reflected by thereflectors 20 to the IDT 12 side with a high reflection coefficient.Then, the stop band upper end mode SAW energy confinement becomesstronger, and it is possible to realize a low-loss resonator.

As opposed to this, in the event that the relationship between the stopband upper end frequency ft2 of the IDT 12 and the stop band lower endfrequency fr1 of the reflectors 20 and stop band upper end frequency fr2of the reflectors 20 is set to the condition of ft2<fr1, or thecondition of fr2<ft2, the reflection coefficient Γ of the reflectors 20at the stop band upper end frequency ft2 of the IDT 12 decreases, and itbecomes difficult to realize a strong energy confinement condition.

Herein, in order to realize the condition of Equation (32), it isnecessary to make a frequency shift of the stop band of the reflectors20 to an area higher than the stop band of the IDT 12. Specifically,this can be realized by making the array cycle of the conductor strips22 of the reflectors 20 shorter than that of the array cycle of theelectrode fingers 18 of the IDT 12. Also, as other methods, it can berealized by making the thickness of the electrode film formed as theconductor strips 22 of the reflectors 20 less than the thickness of theelectrode film formed as the electrode fingers 18 of the IDT 12, or bymaking the depth of the inter-conductor strip grooves of the reflectors20 less than the depth of the inter-electrode finger grooves of the IDT12. Also, a plurality of these methods may be employed in combination.

According to FIG. 50, it can be said that, apart from the high Q, it ispossible to obtain a high figure of merit M. Also, FIG. 51 is a graphshowing the relationship between an impedance Z and the frequency forthe SAW resonator from which FIG. 50 is obtained. From FIG. 51, it canbe seen that no unnecessary spurious response exists in the vicinity ofthe resonance point.

In the heretofore described embodiment, the IDT 12 configuring the SAWresonator 10 is shown in such a way that all the electrode fingersintersect alternately. However, the SAW resonator 10 according to theinvention can also achieve a considerable advantage with only the quartzcrystal substrate thereof. For this reason, even in the event that theelectrode fingers 18 in the IDT 12 are thinned out, it is possible toachieve the same kind of advantage.

Also, regarding the grooves 32 too, they may be partially providedbetween the electrode fingers 18 and between the conductor strips 22 ofthe reflectors 20. In particular, as the central portion of the IDT 12,which has a high oscillatory displacement, has a dominant effect on thefrequency-temperature characteristics, a structure may be adoptedwherein the grooves 32 are provided only in that portion. With this kindof structure too, it is possible to achieve a SAW resonator 10 with goodfrequency-temperature characteristics.

Also, in the heretofore described embodiment, it is noted that Al or anAl-based alloy is used as the electrode film. However, the electrodefilm may be configured using another metal material, provided that it isa metal which can achieve the same advantage as the embodiment.

Also, although the heretofore described embodiment is a one-terminalpair SAW resonator in which only one IDT is provided, the invention isalso applicable to a two-terminal pair SAW resonator in which aplurality of IDTs are provided, and is also applicable to alongitudinally coupled or transversally coupled double mode SAW filteror multiple mode SAW filter.

Next, a description will be given, referring to FIG. 55, of the SAWoscillator according to the invention. The SAW oscillator according tothe invention, as shown in FIG. 55, is configured of the SAW resonator10, an IC (integrated circuit) 50, which applies voltage to, and drivecontrols, the IDT 12 of the SAW resonator 10, and a package which housesthem. In FIG. 55, FIG. 55(A) is a plan view with a lid removed, and FIG.55(B) is a diagram showing a cross-section along A-A in FIG. 55(A).

In the SAW oscillator 100 according to the embodiment, the SAW resonator10 and IC 50 are housed in the same package 56, and electrode patterns54 a to 54 g formed on a bottom plate 56 a of the package 56, and thepectinate electrodes 14 a and 14 b of the SAW resonator 10 and pads 52 ato 52 f of the IC 50, are connected by metal wires 60. Then, a cavity ofthe package 56 housing the SAW resonator 10 and IC 50 is hermeticallysealed with a lid 58. By adopting this kind of configuration, it ispossible to electrically connect the IDT 12 (refer to FIG. 1) the IC 50,and an unshown externally mounted electrode formed on the bottom surfaceof the package 56.

Therefore, in response to a demand for an expansion of operatingtemperature range and higher accuracy of internally mounted electronicdevices, with the effect of internal heat generation increasing alongwith the miniaturization of blade servers and other packages, inaddition to a higher reference clock frequency due to the speeding-up ofinformation communication in recent years, and furthermore, in responseto a market which needs long-term, stable operating in environments fromlow temperature to high temperature, such as wireless base stationsinstalled outdoors, the SAW oscillator according to the invention ispreferred, as it has extremely good frequency-temperaturecharacteristics of a frequency fluctuation amount of approximately 20(ppm) or less in its operating temperature range (service temperaturerange: −40° C. to +85° C.).

Furthermore, as the SAW resonator according to the invention, or SAWoscillator including the SAW resonator, realizes a significantimprovement in frequency-temperature characteristics, it contributeslargely to realizing a product with, as well as extremely goodfrequency-temperature characteristics, excellent jitter characteristicsand phase noise characteristics, for example, a mobile telephone, a harddisc, a personal computer, a tuner receiving a BS and CS broadcast, aninstrument processing a high frequency signal transmitted through acoaxial cable or an optical signal transmitted through an optical cable,or an electronic instrument such as a server network instrument orwireless communication instrument which needs a high frequency, highaccuracy clock (low jitter, low phase noise) in a wide temperaturerange, and it goes without saying that it contributes greatly to furthersystem reliability and quality improvement.

As heretofore described, as the SAW resonator according to the inventionhas inflection points within the operating temperature range (servicetemperature range: −40° C. to +85° C.), as shown in FIG. 48, it ispossible to realize frequency-temperature characteristics which describea tertiary curve, or an approximate tertiary curve, with an extremelysmall frequency fluctuation amount of approximately 20 ppm or less.

FIG. 56(A) is a graph showing the frequency-temperature characteristicsof the SAW resonator disclosed in JP-A-2006-203408. Although thefrequency-temperature characteristics describe a tertiary curve, as aninflection point exists in an area beyond the operating temperaturerange (service temperature range: −40° C. to +85° C.), as can be seen,it is essentially a quadratic curve which has an upwardly convex peak,as shown in FIG. 56(B). For this reason, the frequency fluctuationamount has an extremely high value of 100 (ppm).

As opposed to this, the SAW resonator according to the invention, withthe frequency fluctuation amount describing a tertiary curve, or anapproximate tertiary curve, within the operating temperature range,realizes a dramatic reduction of the frequency fluctuation amount.Changes in the frequency fluctuation amount within the operating rangefor a SAW resonator whose IDT and reflectors are covered in a protectivefilm are shown in FIGS. 57 and 58.

The example shown in FIG. 57 is a diagram showing the frequencyfluctuation amount within the operating temperature range when theelectrodes are coated with alumina as a protective film. According toFIG. 57, it can be seen that it is possible to keep the frequencyfluctuation amount within the operating temperature range at 10 (ppm) orless.

Basic data of SAW resonator according to example shown in FIG. 57

H: (material: aluminum): 2,000 (Å)

G: 4,700 (Å) (H+G=0.067)

IDT line occupation rate ηi: 0.6Reflector line occupation rate ηr: 0.8In-plane rotation ST cut substrate with Euler angles (0°, 123°, and43.5°)Pair number: 120Intersection width: 40λ (λ=10 (μm))Reflector number (per side): 36No electrode finger tilt angleProtective film (alumina) thickness 400 (Å)Secondary temperature coefficient 62 =+0.0007 (ppm/° C.²)

The example shown in FIG. 58 is a diagram showing the frequencyfluctuation amount within the operating temperature range when theelectrodes are coated with SiO₂ as a protective film. According to FIG.58, it can be seen that it is possible to keep the frequency fluctuationamount within the operating temperature range at 20 (ppm) or less.

Basic data of SAW resonator according to example shown in FIG. 58

H: (material: aluminum): 2,000 (Å)

G: 4,700 (Å) (H+G=0.067)

IDT line occupation rate ηi: 0.6Reflector line occupation rate ηr: 0.8In-plane rotation ST cut substrate with Euler angles (0°, 123°, and43.5°)Pair number: 120Intersection width: 40λ (λ=10(μm))Reflector number (per side): 36No electrode finger tilt angleProtective film (SiO₂) thickness 400 (Å)Secondary temperature coefficient β=+0.0039 (ppm/° C.²)

The entire disclosure of Japanese Patent Application No. 2009-045359,filed Feb. 27, 2009 and Japanese Patent Application No. 2009-050112,filed Mar. 4, 2009 are expressly incorporated by reference herein.

1. A surface acoustic wave resonator provided on a quartz crystalsubstrate with Euler angles (−1.5°≦φ≦1.5°, 117°≦θ≦142°, and42.79°≦|Ψ|≦49.57°, comprising: an IDT which excites a stop band upperend mode surface acoustic wave; and inter-electrode finger grooveshollowed out of the substrate positioned between electrode fingersconfiguring the IDT, wherein when the wavelength of the surface acousticwave is λ and the depth of the inter-electrode finger grooves is G, λand G satisfy the relationship of0.01λ≦G and wherein, when the line occupation rate of the IDT is η, theinter-electrode finger groove depth G and line occupation rate η satisfythe relationships of−2.0000×G/λ+0.7200≦η≦−−2.5000×G/λ+0.7775provided that0.0100λ≦G≦0.0500λ−3.5898×G/λ+0.7995≦η≦−2.5000×G/λ+0.7775provided that0.0500λ<G≦0.0695.
 2. The surface acoustic wave resonator according toclaim 1, wherein the inter-electrode finger groove depth G satisfies therelationship of0.01λ≦G≦0.0695λ.
 3. The surface acoustic wave resonator according toclaim 1, wherein when the electrode film thickness of the IDT is H, Hsatisfies the relationship of0<H≦0.035λ.
 4. The surface acoustic wave resonator according to claim 3,wherein the line occupation rate η satisfies the relationship ofη=−1963.05×(G/λ)³+196.28×(G/λ)²−6.53×(G/λ)−135.99×(H/λ)²+5.817×(H/λ)+0.732−99.99×(G/λ)×(H/λ)  (8)
 5. The surface acoustic wave resonator accordingto claim 3, wherein the sum of the inter-electrode finger groove depth Gand electrode film thickness H satisfies the relationship of0.040λ≦G+H.
 6. The surface acoustic wave resonator according to claim 1,wherein Ψ and θ satisfy the relationship ofΨ=1.191×10⁻³×θ³−4.490×10⁻¹×0²+5.646×10¹×0−2.324×10³±1.0
 7. The surfaceacoustic wave resonator according to claim 1, wherein when a stop bandupper end mode frequency in the IDT is ft2, a stop band lower end modefrequency in reflectors disposed in such a way as to sandwich the IDT inthe propagation direction of the surface acoustic wave is fr1, and astop band upper end mode frequency of the reflectors is fr2, ft2, fr1,and fr2 satisfy the relationship offr1<ft2<fr2.
 8. The surface acoustic wave resonator according to claim1, wherein inter-conductor strip grooves are provided between conductorstrips configuring the reflectors, and the depth of the inter-conductorstrip grooves is less than that of the inter-electrode finger grooves.9. A surface acoustic wave oscillator comprising: the surface acousticwave resonator according to claim 1; and an IC for driving the IDT. 10.An electronic instrument comprising the surface acoustic wave resonatoraccording to claim
 1. 11. An electronic instrument comprising thesurface acoustic wave resonator according to claim
 3. 12. An electronicinstrument comprising the surface acoustic wave resonator according toclaim
 4. 13. An electronic instrument comprising the surface acousticwave resonator according to claim
 5. 14. An electronic instrumentcomprising the surface acoustic wave resonator according to claim
 6. 15.An electronic instrument comprising the surface acoustic wave resonatoraccording to claim
 7. 16. An electronic instrument comprising thesurface acoustic wave resonator according to claim
 8. 17. An electronicinstrument comprising the surface acoustic wave resonator according toclaim
 9. 18. The surface acoustic wave resonator according to claim 2,wherein when the electrode film thickness of the IDT is H, H satisfiesthe relationship of0<H≦0.035λ.
 19. The surface acoustic wave resonator according to claim4, wherein the sum of the inter-electrode finger groove depth G andelectrode film thickness H satisfies the relationship of0.0407λ≦G+H.
 20. An electronic instrument comprising the surfaceacoustic wave resonator according to claim 2.